Direct numerical simulation of compressible

flow over a backward-facing step

K. Sengupta, K. K. Q. Zhang and F. Mashayek

Department of Mechanical and Industrial Engineering

University of Illinois at Chicago

G. B. Jacobs

Department of Aerospace Engineering and Engineering Mechanics

San Diego State University

March 25, 2009

Abstract

In this paper, we perform a direct numerical simulation study to investigative the effect of

compressibility on the flow over a backward-facing step. A spectral multidomain method is used

for simulating the compressible flow at Reynolds number of 3000 over an open backward-facing

step. The role of compressibility in the sub-sonic regime is studied by simulating the flow at

different free-stream Mach numbers.

We analyze the effect of compressibility on the averaged flow and the Reynolds stresses.

Results indicate that the effect of compressibility on the separated and reattached shear layer

in the backward-facing step is different from that on spatial and temporal mixing layers studied

1

previously. While the canonical free shear flows exhibit a reduction in growth rate with increase

in Mach number, the shear layer in the present study shows an opposite trend. We compute

the budget of turbulent kinetic energy to determine the plausible causes of such behavior. We

conclude that a greater turbulence production for higher Mach numbers is primarily responsible

for the larger growth rate of the shear layer.

1 Introduction

Separating and reattaching turbulent flows have immense technological implications. They

occur in many practical devices like diffusers, combustors and around airfoils and bluff bodies

to name a few. Flow over a backward-facing step (BFS) is a configuration that is often studied

to understand the physics of separated flows. Moreover, combustors and afterburners in ramjet

engines are modeled after such geometries. The flow dynamics in BFS can be decomposed as

follows: The boundary layer upstream of the step separates at the step corner, becoming a free

shear layer. The free shear layer expands over the step and creates a recirculation region, before

reattaching to the bottom wall. The flow subsequently recovers to an equilibrium boundary

layer profile.

Since the pioneering experimental study by Eaton and Johnston1 there has been consider-

able amount of research into backward-facing step flows. Armaly et al.2 presented experimental

studies on the effect of Reynolds number on the reattachment length for a given expansion ratio.

The flow appeared to be three-dimensional for Reynolds number (based on the step height and

the inlet free stream velocity) above 400. The reattachment length was found to increase with

Reynolds number upto 1200, then decreased in the transitional range 1200 < Re < 6600 and

finally remained relatively constant in the fully turbulent range (Re > 6600). The effects of

expansion ratio on the reattachment length were studied by Durst and Tropea3. The reattach-

2

ment length also depends on the state of the flow at separation and the ratio of the boundary

layer thickness to step height. Adams et al.4 investigated the effect of upstream boundary layer

profile on the reattachment length. The effect of inlet turbulence intensity on the reattachment

process was studied by Isomoto and Honami5.

Numerous computational studies have also been conducted for BFS flow in the past two

decades. Most of the early numerical simulations have been confined to two-dimensional flows.

Kim and Moin6 used a fractional step method to compute incompressible two-dimensional flow.

The discrepencies with the experimental data of Armaly et al.2 were attributed to the onset

of three-dimensionality above Reynolds number 600. Kaikatis et al.7 identified bifurcation of

two-dimensional laminar flow to three-dimensional flow at Rec=700, as the primary source of

mismatch between simulation and experimental data. In a later work8 Kaikatis found that

the unsteadiness was created by convective instabilities. Le et al.9 conducted the first direct

numerical simulation for a three-dimensional backward-facing step configuration for a Reynolds

number (based on step height and free stream velocity) of 5100 and expansion ratio (ratio of

the height of the post expansion section to that of the inlet channel) of 1.2. Their results

were compared with the experiment of Jovic and Driver10. In addition to typical engineering

quantities, such as averaged skin friction and pressure coefficients, Le et al.9 reported on the

unsteady characteristics of the flow. The mean streamwise velocity and Reynolds stresses in

the recirculation, reattachment and recovery regions compared well with the experimental data.

They also computed the budgets of turbulent kinetic energy and Reynolds stresses. Kaltenbach

and Janke11 investigated the effect of ‘sweep’ on transitional separation bubble behind the step.

Sweeping essentially amounts to skewing the inlet velocity profile in the spanwise direction.

In their baseline unswept case at Re=3000, the boundary layer at the step was laminar and

underwent transition before reattaching. A similar configuration was later simulated by Wengle

et al.12 More recently, Barkley et al.13 carried out three-dimensional linear stability analysis for

3

an expansion ratio of 2. They showed that the primary bifurcation of the steady, two-dimensional

flow is a steady, three-dimensional instability.

The computational studies mentioned above have dealt with incompressible flows. Renewed

interest in high speed propulsion has invigorated fundamental research on compressible tur-

bulent flows (see Lele14 for a review). Compressibility effects have been studied in detail for

free shear flows. Stabilizing effect of compressibility has been one of the most distinguish-

ing features of compressible shear flow from its incompressible counterpart. The convective

Mach number Mc, introduced by Bogdanoff15 is the most widely used parameter for determin-

ing compressibility effects. For streams with equal specific heats the convective Mach number,

Mc = (U1 − U2)/(c1 + c2), where U1, U2 are free stream velocities and c1, c2 are the free stream

sound speeds. A number of studies (see e.g. Papamoschu and Roshko16, Hall et al.17, Clemens

and Mungal18) have shown large reduction in growth rate with increasing Mc. Several different

explanations for inhibited growth rate of shear layer have been offered. Zeman19 and Sarkar20

proposed that dilatational dissipation and pressure-dilatation, which acts as sink of turbulent

kinetic energy, could at sufficiently high Mach numbers lead to reduction in turbulence levels

and thereby inhibit shear layer growth. However, the above explanation was later replaced by

alternate theories by Sarkar21 and Vreman et al.22, which indicated that reduced turbulent pro-

duction, and not the dilatation terms, was responsible for decreased turbulent kinetic energy.

A direct relationship between momentum thickness growth rate and the integrated turbulent

production was established for the mixing layer by Vreman et al.22 They further showed that the

reduction in turbulent production is due to reduced pressure fluctuation via the pressure-strain

term. For uniform shear flow, Sarkar23 found that the pressure gradient term in the momentum

equation and the pressure-strain term in the Reynolds stress equation lead to reduced turbulence

levels. Freund et al.24 also confirmed the reduction of the pressure strain term in the annular

mixing layer. In their DNS study, Pantano and Sarkar25, showed that the gradient Mach num-

4

ber, which is the ratio of the acoustic time scale to the flow distortion time scale, is the key

parameter which determines the reduction in pressure-strain term in compressible shear flows.

There has been dearth of LES/DNS studies of compressible separating and reattaching flows.

Avancha and Pletcher26 carried out large-eddy simulation of flow over a backward-facing step

with wall heat flux at single low Mach number of 0.006. The focus of their study was to analyze

the heat transfer characteristics in the flow. Their major conclusions were an occurrence of

peak heat transfer rate slightly upstream of the reattachment, and invalidation of the Reynolds

analogy within the recirculation zone. In a recent study, Dandois et al.27 carried out direct

numerical and large-eddy simulation of flow over a rounded ramp at a single free stream Mach

number 0.3. The motivation of the study was to control the separation using synthetic jets at

different frequencies. Numerical simulation of a compressible mixing layer past an axisymmetric

trailing edge was performed by Simon et al.28. The work focussed on evolution of the turbulence

field through extra strain rates (owing to streamline curvature) and on the unsteady features of

the annular shear layer at a single Mach number of 2.46. A hybrid RANS/LES methodology

called zonal detached eddy simulation (ZDES) was used for computing the flow at a Reynolds

number (based on the diameter of the trailing edge) of 2.9 million. A three-dimensional stability

study of compressible flow over open cavities was conducted by Bres and Colonius29. They

performed both linear stability analysis and direct numerical simulation. The authors concluded

that instability mode properties were by and large unaffected by Mach number over a subsonic

range upto 0.6.

As is seen from a survey of the available literature, interest in DNS and LES of compressible

separated shear flows in practical applications has grown only recently. Moreover, there has

been only few studies of the backward-facing step configuration, none of which attempted to

evaluate the role of compressibility. In this context, the present direct numerical computation of

compressible flow over a backward-facing step is an attempt to improve physical understanding

5

of this benchmark flow by looking at different flow statistics and energy budget. The work seeks

to address open questions such as: How does the mean flow and turbulent stresses change with

free stream Mach number? How does the free-stream Mach number affect the growth of the

shear layer? What are the contributions of various quantities in the budget of turbulent kinetic

energy? Paucity of computational resources restricts our investigation to the low Reynolds

number regime. The paper is organized as follows: First we describe the governing equations

in section 2. In section 3, a brief account of our numerical methodology is provided. The

backward-facing step model is described in section 4. In section 5, we validate our simulation

by comparing the mean flow and second order statistics with experiment. The flow topology

is described in section 6. The budget of turbulent kinetic energy is presented and analyzed in

detail in section 7. Finally we summarize the important findings in section 8.

2 Governing Equations

The governing equations for the compressible and viscous fluid flow are the conservation state-

ments for mass, momentum and energy. They are presented in non-dimensional, conservative

form with Cartesian tensor notation,

∂ρ

∂t+

∂(ρuj)

∂xj

= 0, (1)

∂(ρui)

∂t+

∂(ρuiuj + pδij)

∂xj

=∂σij

∂xj

, (2)

∂(ρe)

∂t+

∂[(ρe + p)uj]

∂xj

= − ∂qj

∂xj

+∂(σijui)

∂xj

. (3)

The total energy, viscous stress tensor and heat flux vector are, respectively, given as,

ρe =p

γ − 1+

1

2ρukuk, (4)

6

σij =1

Re

(∂ui

∂xj

+∂uj

∂xi

− 2

3

∂uk

∂xk

δij

), (5)

qj = − 1

(γ − 1)RePrM2f

∂T

∂xj

. (6)

The Reynolds number Re is based on the reference density ρ∗f , velocity U∗f , length L∗f , and

molecular viscosity µ∗f and is given by Re = ρ∗fU∗f L∗f/µ

∗f . Pr = µ∗fCp/k

∗ is the Prandtl number.

The superscript ∗ denotes dimensional quantities. The above equation set is closed by the

equation of state,

p =ρT

γM2f

, (7)

where Mf is defined as Mf = U∗f /c∗f , with c∗f =

√γRT ∗

f denoting the reference speed of sound.

Here, Tf is the reference temperature. Therefore, by definition Mf is the reference Mach number,

which is taken as 1 implying that our reference velocity is the same as the reference speed

of sound. The flow Mach number, which does not appear explicitly in the non-dimensional

equations is defined as, M = U∗f /c∗, where c∗ now is given by, c∗ =

√γRT ∗. With Mf set to

unity, M essentially becomes the reciprocal of the non-dimensional sound velocity. Moreover,

from the above definitions, M (which is a local quantity in the flow) is related to the local non-

dimensional temperature T ∗/T ∗f . In the simulations presented in this paper, the “free-stream”

M , based on the non-dimensional temperature at the wall, is varied to obtain results at different

Mach numbers. The conservation equations can be cast in the matrix form

∂Q

∂t+

∂F ai

∂xi

− ∂F vi

∂xi

= 0, (8)

where

Q =

ρ

ρu1

ρu2

ρu3

ρe

, (9)

7

F ai =

ρui

ρu1ui + pδi1

ρu2ui + pδi2

ρu3ui + pδi3

(ρe + p)ui

, (10)

F vi =

0

σi1

σi2

σi3

−qi + ukσik

. (11)

Here, Q is the vector of the conserved variables, while F ai and F v

i are the advective and viscous

flux vectors respectively, in the xi direction.

3 Numerical Method

In this work, a staggared grid Chebyshev multidomain method is used for solving the governing

transport equations. Here, we provide a brief description of the method. For a more complete

description, the reader is referred to our previous works30–32. The approximation begins with the

subdivision of the computational domain into non-overlapping hexahedral subdomains/elements.

Each element is then mapped onto a unit hexahedron by an isoparametric transformation using

a linear blending formula. Under the mappings, Eq. (8) becomes

∂Q

∂t+

∂F ai

∂Xi

− ∂F vi

∂Xi

= 0, (12)

where

Q = JQ,

F ai =

∂Xi

∂xj

F aj ,

F vi =

∂Xi

∂xj

F vj . (13)

8

In the above equations, Q and Q represent the solution vectors (Eq. (9)) on the physical and

the mapped grid, respectively. J is the Jacobian of the transformation from the physical space

to the mapped space. The fluxes (Eqs. (10) and (11)) on the two grids are given by F and F ,

while the term ∂Xi

∂xjis the metric of the transformation, xj and Xi being the coordinates of the

physical space and the mapped space, respectively.

Within each sub-domain the solution values, Q, and the fluxes, F i, in Eq. (12) are ap-

proximated on separate grids. The solutions are computed on the Chebyshev-Gauss grid while

the fluxes are computed on the Chebyshev-Gauss-Lobatto grid. In one space dimension, the

Chebyshev-Gauss and Chebyshev-Gauss-Lobatto quadrature points are defined by,

Xj+1/2 =1

2

{1− cos

[(2j + 1)π

2N

]}, j = 0, ....N − 1, (14)

and

Xj =1

2

{1− cos

[πj

N

]}, j = 0, ....N, (15)

respectively, on the unit interval [0, 1]. In three-dimension, the corresponding grids are given by

the tensor product of the above one-dimensional grids. Thus the solution is approximated as,

Q(X, Y, Z) =N−1∑

i=0

N−1∑

j=0

N−1∑

k=0

Qi+1/2,j+1/2,k+1/2hi+1/2(X)hj+1/2(Y )hk+1/2(Z), (16)

where hj+1/2 ∈ PN−1 is the Lagrange interpolating polynomial defined on the Gauss grid.

The computation of the advective (inviscid) and viscous fluxes follow different procedures.

The inviscid fluxes are computed by reconstructing the solution at the Gauss-Lobatto points

using the interpolant (Eq. (16)). This, however, leads to two different flux values at the sub-

domain interface points, one from each of the two contributing sub-domains. In-order to enforce

continuity of the fluxes, Roe’s approximate Riemann solver33 is used to compute a single flux

from the two values. For a detailed description of the role of the Riemann solver, we refer to

Kopriva and Kolias34.

9

The computation of the viscous flux uses a two-step procedure. Since the reconstruction of

the solution from the Gauss points onto the Lobatto points gives a discontinuous solution at the

element faces, it is necessary to construct first a continuous piecewise polynomial approximation

before differentiating. To construct the continuous approximation, the average of the solutions

on either side of the interface is used as the interface value. The continuous solution is then dif-

ferentiated to obtain the derivative quantities needed for the viscous fluxes. Since differentiation

reduces the polynomial order by one, it is natural to evaluate the differentiated quantities on

the Gauss grid. Once the derivative quantities are evaluated at the Gauss points, a polynomial

interpolant of the form (Eq. (16)) is defined, so that the gradients can be evaluated at the Lo-

batto points. From the cell face values, the viscous fluxes are computed and combined with the

inviscid fluxes to obtain the total flux. Finally, a semi-discrete equation for the solution values

is obtained by computing the gradient of the total fluxes at the Gauss points. The equation is

advanced in time using a 4th-order low storage Runge-Kutta scheme.

Note that evaluating the gradients at the cell centers has two desired effects. First, it makes

the evaluation of the divergence consistent with that used in the continuity equation. Second,

the evaluation of the viscous fluxes will not require the use of sub-domain corner points.

4 Computational Model

To the best of the authors’ knowledge there has not been any experiment for compressible flow

over backward-facing step in the Reynolds number regime that is feasible for DNS. In this work

we simulate the backward-facing step geometry investigated by Wengle et al.12 (hereafter referred

to as WHBJ). The WHBJ experiment was conducted for incompressible flow at a single Reynolds

number of 3000. Figure 1 shows the computational domain used in this study. The streamwise

length (L) is 22.92h including an inlet section Li = 5h, where h is the step-height taken as 1. The

10

total height in the z-direction is 11.76. The spanwise extent is taken to be 6h. The co-ordinate

system is placed at the lower left corner of the domain as shown in the figure. The multi-

domain spectral method allows control of resolution at two levels: the number of sub-domains

(h-resolution) and polynomial order within each sub-domain (p-resolution). The number of

sub-domains employed for the inlet section are hx = 8, hy = 12, hz = 6. The distribution of sub-

domains in the x-direction and z-direction conforms to a power-law, while uniform element sizes

are considered in the y-direction. For x > 5, a similar power law distribution with 32 elements is

used in the x-direction. The region behind the step (z < 1) is discretized with 6 sub-domains in

the z-direction having a cosine distribution. The region above the step (z > 1) has 6 sub-domains

with a power law distribution. As with the inlet section, uniform spacing of 12 elements is used

in the y-direction. The above distribution gives finer grid at the step corner, in the shear layer

and in the boundary layer.

4.1 Initial and boundary conditions

At the inlet (x = 0), following Ref.12, a laminar boundary layer profile with δ0.99/h = 0.21 is

imposed. The density is set to ρ∞ = 1.0, while the temperature is set to 1/M2. Pressure is

obtained from the density and temperature using the equation of state (Eq. (7)). At the top

boundary z = 11.76, the primitive flow variables are set to the same freestream values as in the

inflow. The outflow boundary conditions for the multi-domain Chebyshev spectral method have

been studied by Jacobs et al.35. The outflow boundary condition employed here follows the

Boundary SPECified (BSPEC) condition described in Ref.35 The exterior state value for the

streamwise velocity, u, is the average velocity profile extracted from WHBJ. The pressure is set

to its free stream value p∞, with the expectation that at a far distance from the step the flow

will return to its free stream state. Conservative isothermal wall boundary condition of Jacobs

et al.31 is used at the wall with the wall temperature set to 1/M2. The Reynolds number based

11

on the step height and the velocity outside the boundary layer (U∞) at the step is 3000, same

as in WHBJ.

The flow at z > 1 is initialized with the uniform inlet condition while for z < 1 all the

variables are set to zero. This ensures that mass is conserved initially and since the method is

conservative, mass conservation is ensured throughout the computation.

5 Validation and Convergence Study

The purpose of this section is to validate our numerical methodology by comparing with the

measurements of WHBJ. Several cases investigated are listed in table 1. All simulations are

conducted in three stages: (1) establish a stationary turbulence, (2) calculate averages, and (3)

calculate fluctuations and related statistics. The simulation is run for approximately 10 flow-

through times to reach the statistically stationary state. The mean statistics are then computed

for over 5 flow-through times and finally another 10 flow-through times are used to compute the

higher order statistics. The dimensional flow-through time is given by L/U∞, the time required

by a fluid particle to traverse the length of the computational domain. Compressibility effects

are expected to be small for Ma2 (the lowest Mach number case) and therefore, Ma2 will be

treated as the base case for comparison with the incompressible experiment12.

Reattachment length

The size of the mean recirculation vortex is used as an important global measure for deter-

mining the accuracy of backward-facing step computations9;36. The size of the recirculation

region is characterized by the reattachment length, Lr. Reattachment length is the distance

from the step wall to the point of zero wall shear stress (τw = 0 or dU/dz = 0). Its value

depends on various parameters, such as (a) Reynolds number based on the step height, Reh,

(b) state of the flow at the separation, (c) the ratio of boundary layer thickness to step height

12

at the edge of the step, (d) turbulence intensity in the free stream and (e) expansion ratio.

The mean reattachment lengths reported in WHBJ are Lr = 6.4 and Lr = 6.5 for experiment

and computations, respectively. Table 1 shows the mean reattachment length for all the cases

along with the magnitude of deviation from the above experimental value. Clearly, free stream

Mach number affects the reattachment location, which is closer to the step for higher Mach

numbers. Ma2 provides the closest agreement with the experiment. Further reduction in Mach

number (≈ 0.1) leads to severe limitation on the time step resulting in inordinately large total

computational time. Therefore, we restricted the lowest Mach number in our study to 0.2. The

physical origin of the variation in reattachment length with Mach number will be discussed in

Section 7.

Averaged velocity

We undertake a systematic comparison of our simulation results with the averaged velocity

data from the experiment in WHBJ. One of the distinguishing aspects of our spectral multido-

main method is the feature of controlling the spatial resolution at two different levels. The

resolution can be altered either by changing the number of sub-domains (h-refinement) or by

changing the order of the polynomial within each sub-domain (p-refinement). The theory of

h/p convergence37 indicates that p-refinement has exponential convergence while h-refinement

leads to only an algebraic convergence rate. Therefore, we conduct a p-convergence study for

our backward-facing step simulations at Ma=0.4 with three different polynomial orders, p=6,

8, 10, the corresponding cases denoted by Ma4, Ma4-p8 and Ma4-p10 in table 1. Figure 2 plots

the Favre averaged streamwise and wall normal velocities for the different polynomial orders,

in the recirculation (x/H = 3, 4), reattachment (x/H = 5) and recovery region (x/H=6). The

figure indicates that the differences resulting from an increase in polynomial order are small.

Therefore, the mean flow is well resolved with the lowest polynomial order (p=6). At stations

x/H = 4 and x/H = 5 we observe significant differences with experiment values. The differ-

13

ences are amplified for the wall normal velocity, due to their smaller range compared to the

streamwise velocity. The fair agreement in the mean velocities for Ma=0.4 is consistent with

the underprediction of the mean reattachment length (see table 1). The obvious physical differ-

ence between the experiment and our simulations at Ma=0.4 is compressibility. This argument

is supported in figure 3, where the Favre averaged velocities for different Mach numbers are

plotted. Ma2 profiles are in good agreement with the experiment, implying that increasing the

free-stream Mach number can significantly alter the averaged flow field for the backward-facing

step geometry.

Reynolds stresses

In turbulent shear flows, the Reynolds stresses (second-order turbulent statistics) are harder

to predict than the averaged quantities (first-order statistics). As before, we establish p-

convergence for the second-order statistics by analyzing normal and shear stresses for Ma4 and

Ma4-p8. Figure 4 shows the streamwise normal stress and shear stress at different streamwise

locations, downstream of the step. Good agreement between the two cases confirms that p=6

provides adequate resolution even for the second-order statistics. The small cusps in the pro-

files in figure 4 are an artifact of the discontinuity of the solution at the sub-domain interfaces,

which is inherent in our multi-domain spectral method. Since Reynolds stresses are given by the

covariances of the solution variables (velocities), any discontinuity at the interface points will

naturally have more impact on them than on the averaged quantities (figures 2 and 3), where no

cusps appear. It must be noted that lack of smoothness does not imply lack of accuracy. The

discontinuity in the solution across the sub-domains will reduce with increase in the polynomial

order of approximation. This is confirmed in figure 4, where using p=8 results in smoother

profiles than p=6.

WHBJ provides measurements of streamwise normal stress {uu} and shear stress {uw}, which

are compared with our simulation results in figure 5. The separating boundary layer is laminar,

14

confirmed by the low intensity levels (0.3%) just upstream of the step. Figure 5 reveals a

large difference in both normal and shear stresses between Ma2 and the higher Mach number

cases in the recirculation region (x/H = 4). This implies a faster transition to turbulence for the

separated shear layer at higher Mach numbers, resulting in a more turbulent recirculation region.

Differences between the experiment and the base case (Ma2) is consistent with the findings

in WHBJ, where similar differences between experiment and computation were observed. At

x/H = 6, Ma3 and Ma4 seem to agree better with the experiment, but this to the authors

opinion is more fortuitous than physical and can be explained from the streamwise location of

the profiles for each case relative to the reattachment point. The station x/H = 6 is near the

reattachment point for Ma2 (Lr = 6.0), but lies in the early recovery region of the flow for

Ma3 (Lr = 5.45) and Ma4 (Lr = 5.42), while in the experiment the location is still half a step

height away from the reattachment point. As the flow evolves, the turbulent stresses increase

upto the reattachment point and then start to decay downstream of it. Therefore, the Reynolds

stresses for Ma3 and Ma4, representative of the early recovery region are incidentally closer to

the experiment, corresponding to the backflow region, than are the stresses for Ma2. Further

downstream, x/H = 8, stresses for Ma2 are again closer to the experiment than Ma3 and Ma4,

consistent with the overall trend.

6 Flow Topology

Topology of the three-dimensional flow over a backward-facing step is very complex. The flow

structures have been investigated in the studies by Neto et al.36 Figure 6 shows the pressure

iso-contours for the case Ma4. Pressure iso-contours provide a good indicator of various vortical

structures since at the center of a viscous vortex pressure has a minimum and increases radially

outward38. The iso-contour at any radial distance from the pressure minima visualizes the

15

vortex through a closed contour. The figure shows that as the shear layer goes into transition,

two-dimensional Kelvin-Helmhotz billows are created at about 3 step heights downstream of

the step. These spanwise rollers, though nominally two-dimensional, still exhibit waviness in

the spanwise direction. As the shear layer grows, the vortices start interacting (helical pairing)

and transform into staggered arch-like vortices which impact the lower wall at the reattachment

region. The accompanied bursting of these structures upon impact, creates three-dimensional

longitudinal vortices.

The underlying vorticity field in the flow can be visualized using the concept of helicity.

Formally, helicity for a fluid confined to a domain D, is the integrated scalar product of the

three-dimensional velocity and vorticity field39,

H =∫

D

~U · ~ωdV (17)

where ~U and ~ω are the velocity and vorticity, respectively. Physically, helicity can be interpreted

as the extent of “cock screw” like motion present in the flow. Regions of high helicity indicate

that the axis of rotation of the local fluid elements is aligned with the fluid velocity. Therefore,

helical structures are located by identifying vortices that have non zero velocity along their

axes. Figure 7 depicts helical structures represented by vortex cores, colored by the streamwise

velocity. The vortex cores are located by finding cells in which vorticity is aligned with velocity.

Cells in which two or more faces of the cell have this criterion are determined to have been

pierced by the core center-line. Clustering of vortex cores in the shear layer indicates a strong

“cock screw” type motion of the fluid. Qualitative changes in the flow resulting from vortex

breakdown are presumably closely related to large changes in helicity39, which may therefore be

used to characterize such events. Decrease in vortex cores (figure 7) beyond the reattachment

region implies breakdown of the coherent structures as the shear layer impinges on the bottom

wall.

16

7 Analysis of Compressibility Effects

Analysis of the averaged flow field and Reynolds stresses shows that the change in the free-

stream Mach number affects the flow. A shorter reattachment length with increase in Mach

number indicates that the separated shear layer over a backward-facing step is less stable with

increase in compressibility. This behavior is different from free shear layers, where compress-

ibility suppresses the growth rate thereby promoting stability. Before analyzing the probable

physical mechanism responsible for such anomalous behavior, we would like to verify that our

numerical methodology is able to capture the effect of compressibility on a classical free-shear

layer. This verification serves two purposes: first it provides an additional validation of our

methodology and second it would indicate that the opposite trend observed for the shear layer

in the backward-facing step has a physical origin.

Therefore, in the following sub-section we simulate a compressible free shear layer at different

Mach numbers. The rest of the section is devoted to the detailed analysis of the effect of

compressibility on flow over the backward-facing step.

7.1 Two-dimensional compressible free shear layer

Here we asses the ability of our numerical methodology to reproduce the well known effect of

compressibility on free-shear flows. Numerous studies (see25) on compressible shear layers have

established that the growth rate of the shear layer decreases with increase in compressibility,

manifested through an increase of convective Mach number. The convective Mach number, Mac,

introduced by Bogdanoff15 is used to determine compressibility effects in turbulent shear layers.

Mac is defined as,

Mac =U1 − U2

c1 + c2

, (18)

17

where U1 and c1 are the velocity and speed of sound on the high-speed side of the shear layer

while U2 and c2 are the corresponding values on the low speed side.

As a canonical test case we simulate a spatially developing two-dimensional planar shear layer

with our Chebyshev spectral multidomain method. A two-dimensional configuration is chosen

with the understanding that though certain features of three-dimensional turbulence, such as

vortex stretching, would be absent, a two-dimensional shear layer exhibits an inhibited growth

rate with increase in compressibility just like its three-dimensional counterpart40. Previous work

on two-dimensional shear layers (see Lesieur et al.41, Comte et al.42, Sandham and Reynolds43,

Wilson and Demuren44 and Stanley and Sarkar45) was motivated in part to ascertain the ex-

tent to which two-dimensional simulations can capture the experimentally observed behavior of

the growth rate and different statistical moments. Moreover, simulation of a three-dimensional

spatially growing shear layer has steep resolution requirements. To circumvent the large com-

putational cost, three-dimensional simulations are usually of temporal nature (46,25). Though

temporal simulations can capture the dynamics of vortex roll up and pairing, they fail to ac-

count for the effects of the velocity ratio across the layer, divergence of streamlines and spatially

non-uniform convection velocities of various structures. These are effects that are typical of the

shear layer in the backward-facing step.

The flow configuration follows the one investigated by Stanley and Sarkar45. A schematic

of the computational domain is shown in figure 8. The dimensional domain extents are given

by Lx = 143δw(0) and Ly = 38δw(0), where δw(0) = 4U/|∂U/∂y|max is the initial vorticity

thickness. The Reynolds number based on the momentum thickness, Reθ, is 180, while the

free-stream velocities of the two layer gives rise to a velocity ratio, (U1−U2)/(U1 + U2), of 0.33.

At the inflow boundary a hyperbolic tangent shear layer profile,

u =U1 + U2

2+

U1 − U2

2tanh

(y

2θm

)(19)

is set for the streamwise velocity. In the above expression θm is the momentum thickness of the

18

shear layer, while U1 and U2 are the velocities of the high and low-speed streams, respectively.

The lateral velocity is taken as zero. Density and temperature are set to their uniform free-

stream values. The streamwise velocity at the top and bottom is set to U1 and U2, respectively.

The inflow values for the primitive variables are applied at the outflow boundary. The flow is

initialized with the hyperbolic-tangent profile (equation 19) for the streamwise velocity, while

density and temperature is taken to be their free-stream values.

In-order to investigate the effect of compressibility, we simulate three cases having Mach

number, Ma = 0.25, 0.5, 0.6. The lowest Mach number case is the “naturally developing” case

in Stanley and Sarkar45. The number of sub-domains employed in the x and y co-ordinate

directions are 32 and 8, respectively, while a polynomial of 7 is used within each sub-domain.

This gives a total of 20,736 Lobatto grid points.

In figure 9 we compare profiles of mean longitudinal velocity (in self similar coordinates)

with the results of Stanley and Sarkar. Good agreement with the published data is observed.

Moreover, the profiles exhibit good degree of self similarity. The growth of vorticity thickness of

the shear layer is plotted in figure 10, where a reduction in the growth rate with Mach number

is readily observed. For Ma=0.25, a linear growth rate is recovered downstream of x > 250.

The corresponding locations for Ma=0.5 and 0.6 are x > 300 and x > 350, respectively.

The results above indicate that our numerical methodology is capable of predicting the inhib-

ited growth rate of free shear layers with increase in compressibility. Therefore, we now return

to the analysis of the flow over backward-facing step, to determine the plausible mechanisms

responsible for the anomalous behavior.

19

7.2 Role of convective Mach number and density ratio

The influence of the free-stream Mach number could manifest on the flow through the change in

convective Mach number and density ratio of the separated shear layer. For the developing shear

layer over the backward-facing step, prescription of U2 and c2 in equation 18, is non-trivial, due

to inhomogeneity in the streamwise direction and the presence of the recirculating region having

negative velocities. From figure 3 we observe that the magnitude of the negative velocities within

the recirculation bubble are small compared to to the free-stream flow. Hence, the separated

shear layer can be considered as a single stream shear layer and the convective Mach number is

determined with U2 ≈ 0. The speed of sound for the low speed stream is calculated based on

the minimum pressure and density, for a given streamwise location. This gives Mac=0.1, 0.15

and 0.2 for Ma2, Ma3 and Ma4, respectively. The above values indicate that the convective

Mach numbers for all three cases is in the quasi-incompressible regime25 and below the critical

value of 0.5 above which significant reduction in growth rate with the convective Mach number

is observed for canonical free shear flows. Therefore, compressibility effects owing to increase in

convective Mach number is expected to be minimal for the present study. Moreover, even if the

convective Mach number were to affect the shear layer development, an increase in its value would

result in slower growth rate, as observed in section 7.1, and consequently a longer reattachment

length. Thus, the increase in convective Mach number does not explain the observed shorter

reattachment for higher free-stream Mach numbers. Clearly, the developing compressible shear

layer in a backward-facing step flow does not follow the trend exhibited by simple free shear

layers.

Next, we investigate the mean density ratio that could influence the development of the shear

layer. Brown and Roshko47 studied the effect of the density ratio across a coflowing shear layer

20

on the growth rate. An expression for spatial growth rate was proposed by Brown48,

dδ

dx= Cδ

(1− r)(1 + s1/2)

2(1 + rs1/2), (20)

where r = u2/u1 denotes the velocity ratio and s = ρ2/ρ1 is the density ratio. The subscripts 1

and 2 refer to the high speed and low speed streams, respectively. In the above expression, δ is

the thickness of the mixing layer, and Cδ is a constant of proportionality. In their study, Brown

and Roshko47 concluded that the shear layer growth rate shows an increase with the increase in

density ratio. The above expression was proposed for a plane mixing layer. Extrapolation of the

relation to the separated and curving shear layer of the present case should be performed with

caution. Figure 11 shows the density profiles in the recirculation region (x/H = 3, 4, 5) and

early recovery region (x/H = 7). The ratio of the density of the free stream flow to the density

of the recirculating flow (ρ1/ρ2) increases with the Mach number. At x/H = 4, the values for

Ma2, Ma3 and Ma4 are 1.008, 1.019 and 1.034, respectively. It was conjectured in the study by

Forliti and Strykowski49 that positive density and velocity gradients with respect to an increased

radial distance from the center of the curved shear layer has a stabilizing effect. The more stable

shear layer is likely to reattach farther from the step, leading to a larger recirculation length.

Therefore, an increased density ratio would in fact result in a larger recirculation bubble for a

higher Mach number, contrary to the observed behavior in our simulations. In summary, both

convective Mach number and density ratio effects are small and do not explain the decrease in

the mean reattachment length with the increase in free stream Mach number.

7.3 Evolution of turbulent kinetic energy and normal Reynolds stresses

Previous study on compressible turbulent shear flows21 has shown that the growth rate of

the mixing layer thickness is proportional to the growth rate of the turbulent kinetic energy.

Figure 12 shows the evolution of the turbulent kinetic energy upto x/H = 5 for Ma2, Ma3

21

and Ma4. Comparison of the three cases reveals that the spatial growth rate increases with

the increase in the free-stream Mach number. Ma2 has significantly lower turbulent kinetic

energy for x/H ≤ 4.5 compared to Ma3 and Ma4. At x/H = 3, Ma4 has a peak value of

0.05 while the peak for Ma2 is only ∼ 0.0125 (75% lower). The difference between Ma3 and

Ma4 is less significant. The increase in turbulent kinetic energy with Mach number for all

locations, x/H ≤ 4.5, implies that the separated shear layer goes into transition earlier and

thereafter grows faster as the free-stream Mach number is increased. The profiles also indicate

that for all cases, as the shear layer develops the location of the peak shifts closer to the bottom

wall, indicating curving of the shear layer. Cross-stream integrated kinetic energy for the same

locations is plotted in figure 13. The figure again underscores that upto x/H = 4.5, the growth

rate increases with Mach number. For Ma3 and Ma4 the integrated turbulent kinetic energy

shows a slight decrease after x/H = 4.5. Therefore, a faster streamwise growth of turbulent

kinetic energy explains why the shear layer grows faster and reattaches earlier with increase in

Mach number, resulting in a shorter mean reattachment length.

Evolution of the peak normal stresses, shown in figure 14, indicates that the streamwise

normal stress, {uu}, is the largest component for all the cases. The streamwise evolution,

however, is similar for all the components. All the stress components increase till approximately

one step-height from the reattachment point followed by a gradual reduction as the shear layer

starts to reattach. Subsequently, in the recovery region the stresses show a rapid decay. This is

unlike the compressible mixing layer past an axisymmetric trailing edge, investigated recently

by Simon et al.28. A distinctly different evolution for streamwise and wall normal stresses was

observed in the above study. Comparison of the different cases in figure 14 shows that there

is an increase in the magnitude of the stresses with the increase in Mach number, during the

development of the shear layer. Consistent with the faster growth of the shear layer, the normal

stresses increase rapidly for Ma3 and Ma4, compared to Ma2. However, as the shear layer relaxes

22

after reattachment the decay of the stresses are identical for the different Mach numbers.

The structural changes of turbulence in shear flows are often investigated by looking into

Reynolds stress anisotropy22;28,

bij =Rij − (2/3)kδij

2k, (21)

where Rij and k are the Reynolds stress and turbulent kinetic energy, respectively. The normal

stress anisotropy is shown in figure 15 at x/H = 4. The anisotropy of streamwise, {uu}, and

wall normal, {ww}, stresses is larger in the region close to the wall. Whereas, the spanwise

stress, {vv}, exhibits more anisotropy in the shear layer region, z/H > 1. On the low-speed side

of the shear layer and in the backflow region (0.125 < z/H < 0.75), the anisotropy in all three

stress components increases with the increase in Mach number. The maximum absolute values

in the region 0.125 < z/H < 0.75 for buu, bvv and bww are 0.16, 0.11 and 0.12, respectively for

Ma2. The corresponding values for Ma3 are 0.05, 0.07 and 0.08, while for Ma4 they are 0.03,

0.05 and 0.07.

7.4 Budget of turbulent kinetic energy: production

The observed difference in the growth rate of the turbulent kinetic energy can be explained by

computing the turbulent kinetic energy budget. In order to derive the transport equation for

turbulent kinetic energy, the Favre decomposition is applied to the convective terms, so that

they reduce to the same form as in the incompressible case. For all the other terms, we apply

the Reynolds decomposition. The derivation technique follows the work of Huang et al.50 and

results in an equation containing a mix of Favre-averaged (curly brackets) and Reynolds-averaged

(angled brackets) quantities,

∂〈ρ〉{uk}{k}∂xk

= −〈ρ〉{u′′i u′′k}∂{ui}∂xk

− ∂〈ρ〉{u′′kk′′}∂xk

− ∂〈p′u′k〉∂xk

23

+∂〈τ ′iku′i〉

∂xk

−⟨

τ ′ik∂u′i∂xk

⟩− 〈u′′k〉

∂〈p〉∂xk

+ 〈u′′i 〉∂〈τik〉∂xk

+

⟨p′

∂u′k∂xk

⟩, (22)

The first term on the right-hand side (RHS) of Eq. (22) represents the energy production;

the second, turbulent diffusion; the third, diffusion resulting from velocity-pressure interaction;

the fourth, viscous diffusion; the fifth, energy dissipation; and the last three are compressibility

related terms, the last one being the pressure dilatation. We demonstrate the accuracy of the

budget calculation, by plotting the left-hand side (LHS) and the RHS of the above equation at

two different streamwise locations. Figure 16 shows that our simulation leads to a good balance

between the two sides of Eq. (22).

Vreman et al.22 showed that the growth rate of a temporal mixing layer is proportional to the

integrated turbulence production. The growth rate of the momentum thickness (δ) was given

by,

dδ

dt= − 2

ρ∞(4U)2

∫ ({ρuw}∂U

∂z

)dz, (23)

where the term within the paranthesis is the turbulence production term. Moreover, the growth

rate of a self-similar spatial mixing layer was shown to be related to the temporal rate in a

transformed co-ordinate. Analysis of the data from the above study revealed that decreased

turbulent production was responsible for the reduced thickness of the shear layer with increase

in Mach number. Vreman et al.22 argued that the reduction in turbulence production with

increase in compressibility was linked to reduced pressure fluctuations via the reduction in the

pressure-strain term. The profiles of turbulent production in the recirculation zone are presented

in figure 17. It is seen that the production increases with the increase in Mach number upto

x/H = 3.5. At x/H = 4 and x/H = 4.5 cases Ma3 and Ma4 have similar profiles, while the

values are still lower for Ma2. An opposite trend with change in Mach number is observed at

x/H = 5.

As the shear layer impinges on the bottom wall in the reattachment region, the vortical

24

structures break up and this leads to a decrease in turbulence production. This is evident in

figure 18, which shows the production at two locations in the reattachment region. We notice

a considerable reduction in production, especially at x/H = 6.5, over that in the recirculation

region. The decrease also explains why a different trend is observed at x/H = 5 (figure 17) as

compared to locations upstream. Since, x/H = 5 is close to the reattachment point for Ma3

and Ma4, the production begins to decay, unlike in Ma2 where it continues to grow, relative to

the locations upstream.

Figure 19 shows the streamwise evolution of the cross-stream integrated turbulence produc-

tion for the three cases. Generally, upto x/H = 4.5, the integrated production shows an increase

with increase in the Mach number. For Ma4 a larger growth rate, compared to Ma2 and Ma3,

is observed upto x/H = 3.5 after which the growth is more gradual. Ma3 similarly exhibits

a higher growth than Ma2 till x/H = 4. This difference in growth of production with Mach

number can be possibly explained by looking at the Reynolds stress component, {uw}, which in

combination with the velocity gradient, ∂U/∂z, is the most significant production term. Profiles

of {uw} and the production at different streamwise locations are shown in figure 20 for Ma2 and

Ma4. The profiles indicate a strong spatial correlation between {uw} and production: the peak

location of production corresponds to the peak location of {uw}. Comparison of the shear stress

profiles for the three cases reveals that the stress increases with increase in the Mach number,

causing the larger production of turbulence thereof. Figure 21 shows the streamwise evolution

of cross-stream integrated shear stress. The plot indicates that upto x/H = 4.5 the integrated

values of shear stress, {uw}, scales with the Mach number.

25

7.5 Budget of turbulent kinetic energy: dissipation and pressure

dilatation

Dilatational dissipation and pressure dilatation are sink terms in the turbulent kinetic energy

equation that can be directly related to compressibility. When fluctuations in viscosity are

neglected, the total dissipation can be decomposed into solenoidal, εs, dilatational, εd, and

inhomogeneous, εI , terms

ε = εs + εd + εI , (24)

where

εs =2

Re〈ω′ijω′ij〉, (25)

εd =4

3

1

Re

⟨∂u′l∂xl

∂u′k∂xk

⟩, (26)

εI =2

Re

(∂2〈u′iu′j〉∂xi∂xj

− 2∂

∂xi

⟨u′i

∂u′j∂xj

⟩). (27)

Previous studies on compressible mixing layers22;25 showed that the dilatational part of the

dissipation was negligibly small and the pressure-dilatation, though not insignificant, was still

small compared to the total dissipation. Neither quantity was used to explain the observed

variation of shear layer growth rate with compressibility. Figures 22 and 23, respectively, show

the solenoidal and dilatational dissipation for the three cases in the recirculation zone. Com-

parison of the two figures reveals that the dilatational dissipation is two orders of magnitude

lower than the solenoidal dissipation for Ma3 and Ma4 and three orders of magnitude lower

for Ma2, and therefore has a negligible contribution to the turbulent kinetic energy budget.

An increase of dilatational dissipation with Mach number is the result of the increase in the

dilatation term (∂u‘k/∂xk) with compressibility. Solenoidal dissipation is therefore the primary

dissipation mechanism in the flow and as seen in figure 22, it is also strongly dependent on

the Mach number, especially in the early stages of shear layer development. Till x/H = 4, εs

26

increases with the increase in the Mach number. Ma2 has significantly lower dissipation than

Ma3 and Ma4. At x/H = 5, the values decrease with increase in the Mach number.

In compressible shear flows the dilatational dissipation is associated with the acoustic and the

entropy mode, while the solenoidal dissipation is associated with the vortical mode51. Thus, the

influence of Mach number on the solenoidal dissipation is expected to be small. The counter-

intuitive results observed in figure 22 therefore warrants some explanation. Budget of solenoidal

dissipation for compressible temporal mixing layers was computed by Kreuzinger et al.52 The

exact transport equation for solenoidal dissipation in compressible flow was considered in their

study. Two configurations, namely, channel flow and mixing layer were studied and the effects

of convective Mach number and mean density variation were investigated. The results indicated

that a change in Mach number did alter the budget of the solenoidal dissipation. However, the

authors concluded that compressibility effects in the transport equation for εs was of indirect

nature. The terms explicitly dependent on compressibility were small for all the cases studied,

except low Mach number mixing layer with large density variation. They argued that the

compressibility effects are proportional to the turbulent Mach number (urms/c) and the gradient

Mach number, both of which are related to the large scales of the flow, and not to the small

scales that determine the dissipation rate, thus explaining the indirect nature of the dependence.

The inhomogenous dissipation, εI , has negligible contribution to the turbulent kinetic energy

budget and therefore is not shown here.

Figure 24 shows the pressure dilatation in the recirculation region. Unlike dilatational dissi-

pation, pressure dilatation is comparable to the solenoidal dissipation (figure 22). As the shear

layer develops, the quantity increases upto locations close to the reattachment point. Thus for

Ma3 and Ma4 the increase is observed upto x/H = 4.5, while for Ma2 the growth is sustained till

x/H = 5. Similarly to dissipation, pressure dilatation is also strongly dependent on the Mach

number. Ma3 and Ma4 have significantly larger values compared to Ma2, the values being one

27

order of magnitude higher at locations close to the step (x/H ≤ 3.5). However, downstream of

x/H = 3.5, there is a large increase for Ma2 resulting in values comparable to the higher Mach

number cases. Thus even at the lowest Mach number studied, explicit effects of compressibility

via pressure-dilatation is non-negligible. This creates a fundamental physical difference between

our compressible flow simulation with the incompressible flow experiment of WHBJ. The dis-

crepancies between the experiment and Ma2, seen in section 5 can now be interpreted in the

light of the above finding.

As the shear layer reattaches, the pressure-dilatation becomes less sensitive to Mach number

change, as seen in figure 25. The solenoidal dissipation on the other hand has higher values

for lower Mach numbers, a trend that was observed earlier in figure 22(f). Such change in the

dependence of solenoidal dissipation on the Mach number indicates that the decay in dissipation

near the reattachment point is more pronounced when compressibility effects are larger.

7.6 Budget of turbulent kinetic energy: turbulent transport quanti-

ties

Turbulent diffusion in the recirculation and reattachment regions is plotted in figure 26. We

observe that the cross-stream profiles have distinct crest and trough. The negative values at the

bottom half of the shear layer indicate that energy is removed from the region, and transported

to the near wall region and to the top half of the shear layer. Comparison of the profiles at

different stations for the three cases, indicates that as the shear layer grows, the region over which

turbulent diffusion occurs widens in the cross-stream direction. At locations, x/H = 3.5 and

4.0, Ma3 and Ma4 have more turbulent transport, than does Ma2. Thereafter, the dependence

on Mach number is weak. The reattachment of the shear layer is accompanied by a reduction

in the turbulent transport (figure 26 (e), (f)).

28

Figure 27 shows the diffusion due to pressure-velocity correlation, in the recirculation and

reattachment regions. In turbulence modeling of compressible flow, the usual practice is to

neglect the pressure diffusion term53. The results here indicate that the magnitude of the term

is comparable to turbulent diffusion transport, and therefore its omission from modeling can

lead to significant error. Unlike turbulent diffusion, pressure diffusion is mostly negative across

the shear layer in the recirculation region (figure 27 (a)-(d)). Moreover, though the region

over which it has significant contribution to the turbulent kinetic energy budget increases with

streamwise distance, the peak value drops. In the reattachment region (x/H=5.5) (figure 27

(e)) there is a change from negative diffusion to positive diffusion.

The remaining terms in Eq. (22), namely, viscous diffusion (term 4), term 6 and term 7

have negligible contributions to the turbulent kinetic energy budget of the flow. Analysis of the

production, sink and transport terms, established production as the single largest contributor

to the budget. The growth of the shear layer and turbulent kinetic energy is therefore strongly

dependent on the production. As a result, higher production closer to the step with the increase

in Mach number leads to a faster development of the shear layer, which consequently reattaches

early and therefore leads to a shorter recirculation vortex.

8 Conclusions

We have investigated compressible flow over a backward-facing step using direct numerical sim-

ulation. In the absence of any compressible flow experiment at low Reynolds numbers (feasible

for DNS), we compare our results with the incompressible experiment of Wengle et al.12. Both

averaged flow and turbulent stresses are found to be sensitive to the free-stream Mach number.

The mean reattachment length decreases with increase in Mach number.

The complex topology of the flow in visualized using pressure iso-contours and helicity. As the

29

upstream laminar boundary layer undergoes transition after separation, Kelvin-Helmhotz billows

appear approximately 3-step heights from the step edge. These structures though nominally

two-dimensional, show waviness in the spanwise direction. Helicity is utilized to determine the

extent of “cockscrew” type motion present in the flow. Clustering of helical structures or vortex

cores within the shear layer indicates a strong “cockscrew” type motion. Subsequent reduction

in density of vortex cores with shear layer reattachment, implies breakdown of the coherent

structures upon impact with the wall.

The decrease in the mean reattachment length with compressibility implies that the separated

shear layer is less stable or grows faster for higher Mach numbers. This behavior is very different

from classical free-shear layers, where compressibility inhibits growth rate. As an aside, we

conduct simulation of a two-dimensional shear layer for different Mach numbers to establish

that our methodology is able to predict the above classical nature of compressible mixing layers.

Finally, we analyze in detail the effect of compressibility on the backward-facing step geome-

try. Effect of convective Mach number and the density ratio are small and does not explain the

anomalous behavior. Growth of mixing layers is linked to evolution of turbulent kinetic energy.

A larger growth in turbulent kinetic energy with Mach number therefore explains the faster

spread of the shear layer. The full budget of turbulent kinetic energy is computed to determine

the effect of compressibility on each of the budget terms. Production is the most significant

term in the budget, which increases with Mach number, thereby resulting in faster growth of

turbulent kinetic energy. Further analysis reveals that the shear stress, {uw}, is responsible

for larger production with increase in compressibility. In the turbulent kinetic energy budget,

dilatational dissipation and pressure dilatation are the terms directly related to compressibility

(dilatation). As in case of canonical mixing layers, dilatational dissipation is negligibly small

for the backward-facing step flow. The primary dissipation therefore comes from the solenoidal

velocity field. Pressure dilatation however, is comparable to the total dissipation, unlike in

30

the canonical free-shear flows investigated before. Among the transport terms, only turbulent

diffusion and diffusion due to velocity-pressure correlation have significant contribution to the

budget. RANS modeling for compressible flows usually neglects the transport due to pressure-

velocity correlation. Our results therefore, underscores that such an omission could lead to large

errors. Turbulent diffusion transports energy from the bottom half of the shear layer to the near

wall and upper half of the shear layer.

The analysis and conclusions presented in this paper establish that the compressible shear

layer over backward-facing step is distinct in its response to increase in compressibility. The

findings can be useful in the design of combustors, flame-holders in high-speed propulsion which

are often modeled after the backward-facing step geometry.

31

References

[1] J. K. Eaton and J. P. Johnston. Turbulent flow reattachment: A experimental study of

the flow structure behind a backward facing step. Report, Thermosciences Division, Dept.

Mechanical Engineering MD-39, Stanford University, 1980.

[2] B. F. Armaly, F. Durst, J. C. F. Peireira, and B. Schonung. Experimental and theoretical

inspection of backward-facing step flow. Journal of Fluid Mechanics, 127:473–496, 1983.

[3] F. Durst and C. Tropea. In Proceedings of 3rd Intl. Symp. on Turbulent Shear Flows,

page 18, Davis, CA.

[4] E. W. Adams, J. P. Johnston, and J. K. Eaton. Experiments on the structure of turbulent

reattaching flows. Report, Thermosciences Division, Dept. Mechanical Engineering MD-43,

Stanford University, 1984.

[5] K. Ishomoto and S. Honami. The effect of inlet turbulence intensity on the reattachment

preocess over a backward-facing step. ASME Journal of Fluids Engg, 111:87–92, 1989.

[6] J. Kim and P. Moin. Application of a fractional step method to incompressible Navier-

Stokes equations. Journal of Comput. Physics, 59:308–323, 1985.

[7] L. Kaikatis, G. E Karniadakis, and S. A. Orszag. Onset of tree dimensionality, equilibria,

and early transition in flow over a backward-facing step. Journal of Fluid Mechanics,

231:501–528, 1991.

[8] L. Kaikatis, G. E Karniadakis, and S. A. Orszag. Unsteadiness and convective instabilities in

a two-dimensional flow over a backward-facing step. Journal of Fluid Mechanics, 321:157–

187, 1996.

[9] H. Le, P. Moin, and J. Kim. Direct numerical simulation of turbulent flow over a backward-

facing step. Journal of Fluid Mechanics, 330:349–374, 1997.

32

[10] S. Jovic and D. M. Driver. Backward-facing step measurement at low reynolds number,

reh=5000. NASA Tech. Mem. 108807, NASA, 1994.

[11] H. J. Kaltenbach and G. Janke. Direct numerical simulation of flow separation behind a

rearward-facing step at Re=3000. Physics of Fluids, 12:2320–2337.

[12] H. Wengle, A. Huppertz, G. Barwolff, and G. Janke. The manipulated transitional

backward-facing step flow: An experimental and direct numerical simulation investigation.

Eur. J. Mech. B - Fluids, 20:25–46, 2001.

[13] D. Barkley, M. G. M. Gomes, and R. D. Henderson. Three-dimensional instability in flow

over a backward-facing step. Journal of Fluid Mechanics, 473:167–190, 2002.

[14] S. K. Lele. Compressibility effects on turbulence. Annual Reviews in Fluid Mechanics,

26:211–254, 1994.

[15] D. Bogdanoff. Compressibility effects in turbulent shear layers. AIAA J, 21:926–927, 1983.

[16] D. Papamoschou and A. Roshko. The compressible turbulent shear layer: An experimental

study. J. Fluid Mech., 197:453–477, 1988.

[17] J. L. Hall, P. E. Dimotakis, and H. Rosemann. Experiments in non-reacting compressible

shear layers. AIAA J., 31:2247–2254, 1993.

[18] J. L. Hall, P. E. Dimotakis, and H. Rosemann. Large-scale structure and entrainment in

the supersonic mixing layer. Journal of Fluid Mechanics, 284:171–216, 1995.

[19] O. Zeman. Dilatational dissipation-the concept and application in modeling compressible

mixing layers. Phys. Fluids, 2:178–188, 1990.

[20] S. Sarkar. The pressure-dilatation correlation in compressible flows. Phys. Fluids,

4(12):2674–2682, 1992.

33

[21] S. Sarkar. The stabilizing effect of compressibility in turbulent shear flow. Journal of Fluid

Mechanics, 282:163–186, 1995.

[22] A. W. Vreman, N. D. Sandham, and K. H. Luo. Compressible mixing layer growth rate

and turbulence characteristics. Journal of Fluid Mechanics, 320:235–258, 1996.

[23] S. Sarkar. On density and pressure fluctuations in uniformly sheared compressible flow. In

IUTAM Symp. on Variable Density Low-Speed Flows, Marseille, France, 1996.

[24] J. B. Freund, S. K. Lele, and P. Moin. Compressibility effects in a turbulent annular mixing

layer. part 1. Turbulence and growth rate. Journal of Fluid Mechanics, 421:229–267, 2000.

[25] C. Pantano and S. Sarkar. A study of compressibility effects in the high-speed turbulent

shear layer using direct simulation. Journal of Fluid Mechanics, 451:329–371, 2002.

[26] R. V. R. Avancha and R. H. Pletcher. Large-eddy simulation of the turbulent flow past

a backward-facing step with heat transfer nd property variation. International Journal of

Heat and Fluid Flow, 23:601–614, 2002.

[27] J. Dandois, E. Garnier, and P. Sagaut. Numerial simulation of active separation control by

a synthetic jet. Journal of Fluid Mech., 574:25–58, 2007.

[28] F. Simon, S. Deck, P. Gullien, P. Sagaut, and A. Merlen. Numerial simulation of the

compressible mixing layer past an axisymmetric trailing edge. Journal of Fluid Mech.,

591:215–253, 2007.

[29] G. A. Bres and T. Colonius. Three-dimensional instabilities in compressible flow over

cavities. Journal of Fluid Mech., 599:309–339, 2008.

[30] G. B. Jacobs, D. A. Kopriva, and F. Mashayek. Validation study of a multidomain spectral

code for simulation of turbulent flows. AIAA Journal, 43(6):1256–1264, 2005.

34

[31] G. B. Jacobs, D. A. Kopriva, and F. Mashayek. A conservative isothermal wall boundary

condition for the compressible Navier-Stokes equation. Journal of Scientific Computing,

30(2):177–192, 2007.

[32] K. Sengupta, G. B. Jacobs, and F. Mashayek. Large-eddy simulation of compressible flows

using a spectral multidomain method. International Journal for Numerical Methods in

Fluids.

[33] P. L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J.

Comp. Phys., 43:357–372, 1981.

[34] D. A. Kopriva and J. H. Kolias. A conservative staggared-grid Chebyshev multidomain

method for compressible flows. Journal of Comput. Physics, 125:244–261, 1996.

[35] G. B. Jacobs, D. A. Kopriva, and F. Mashayek. A comparison of outflow boundary condi-

tions for the multidomain staggered-grid spectral method. Numer. Heat Transfer, Part B,

44(3):225–251, 2003.

[36] A. S. Neto, D. Grand, O. Metais, and M. Lesieur. A numerical investigation of the coherent

vortices in turbulence behind a backward-facing step. Journal of Fluid Mechanics, 256:1–25,

1993.

[37] G. E. M. Karniadakis and S. Sherwin. Spectral/hp Element Methods for Computational

Fluid Dynamics. Oxford University Press, New York, USA, 2005.

[38] G. B. Jacobs. Numerical Simulation of Two-Phase Turbulent Compressible Flows With A

Multidomain Spectral Method. Ph.D. Thesis, University of Illinois at Chicago, Chicago, IL,

2003.

[39] H. K. Moffatt and A. Tsinober. Helicity in laminar and turbulent flow. Annual Review of

Fluid Mechanics, 24:281–312, 1992.

35

[40] M. Lesieur. Turbulence in Fluids. Kluwer Academic Publisher, Dordrecht, Netherlands,

1990.

[41] M. Lesieur, C Staquet, P. L. Roy, and P. Comte. Mixing layer and its coherence examined

from the point of view of two-dimensional turbulence. Journal of Fluid Mechanics, 192:511–

534, 1988.

[42] P. Comte, M. Lesieur, H. Laroche, and X. Normand. In In Turbulent Shear Flows 6, page

361, Berlin.

[43] N. D. Sandham and W. C. Reynolds. In In Turbulent Shear Flows 6, page 441, Berlin.

[44] R. V. Wilson and A. O. Demuren. Numerical simulation of two-dimensional spatially

developing mixing layers. Icase Report 94-32, Institute for Computer Application in Science

and Engineering, NASA Langley Research Center, Hampton, VA, 1994.

[45] S. Stanley and S. Sarkar. Simulations of spatially developing two-dimensional shear layers

and jets. Theoretical Comput. Fluid Dynamics, 9:121–147, 1997.

[46] M. M. Rogers and R. D. Moser. Direct simulation of self-similar turbulent mixing layer.

Physics of Fluids, 6:903–923.

[47] G. L. Brown and A. Roshko. On density effects and large structure in turbulent mixing

layers. J. Fluid Mech., 64:775–816, 1974.

[48] G. Brown. The entrainment and large structure in turbulent mixing layers. In 5th Aus-

tralasian Conf. on Hydraulics and Fluid Mechanics, Canterbury, 1974. Canterbury Univer-

sity.

[49] D. J. Forliti and P. J. Strykowski. An overview of the turbulent shear layer: mixing

processes in propulsion systems. Internal report, dept. mechanical engineering, University

of Minnesota, 2003.

36

[50] P. G. Huang, G. N. Coleman, and P. Bradshaw. Compressible turbulent channel flows: Dns

results and modelling. Journal of Fluid Mechanics, 305:185–218, 1995.

[51] A. J. Smits and J-P. Dussage. Turbulent Shear Layers in Supersonic Flow. Springer,

Springer, 2006.

[52] J. Kreuzinger, R. Friedrich, and T. B. Gatski. A compressibility effect in the solenoidal

dissipation rate equation: A priori analysis and modeling. International Journal of Heat

and Fluid Flow, 27:696–706, 2006.

[53] David C. Wilcox. Turbulence Modeling for CFD. DCW Industries, Inc., La Canada, CA,

1993.

37

List of Tables

1 Cases investigated for the compressible flow over backward-facing step. . . . . . 39

38

Table 1: Cases investigated for the compressible flow over backward-facing step.

Case h p M Lr 4Lr(%)

Ma2 5184 6 0.2 6.0 −6.25

Ma3 5184 6 0.3 5.45 −14.8

Ma4 5184 6 0.4 5.42 −15.3

Ma4-p8 5184 8 0.4 5.30 −17.18

Ma4-p10 5184 10 0.4 5.36 −16.25

39

List of Figures

1 Schematic of the backward-facing step configuration . . . . . . . . . . . . . . . . . . 42

2 Convergence study through comparison of averaged velocities. Streamwise velocity

(top), wall normal velocity (bottom) for Ma=0.4 for different polynomial approximation

orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Comparison of averaged streamwise velocity (top), and wall normal velocity (bottom)

with the experiment of incompressible flow12. . . . . . . . . . . . . . . . . . . . . . 43

4 Convergence study through comparison of Favre averaged stresses for Ma=0.4 for dif-

ferent polynomial approximation orders. . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Comparison of Favre averaged streamwise normal stress (top), and shear stress (bottom)

with the experiment of incompressible flow12. . . . . . . . . . . . . . . . . . . . . . 44

6 Pressure iso-contours indicating the flow topology for the case Ma4. . . . . . . . . . . 44

7 Vortex cores depicting the helical density in the flow for the case Ma4. . . . . . . . . 45

8 Schematic of the free shear layer configuration. . . . . . . . . . . . . . . . . . . . . 45

9 Mean streamwise velocity in self similar co-ordinates for two-dimensional free shear layer. 45

10 Growth of vorticity thickness of two-dimensional shear layer for different Mach numbers. 46

11 Averaged density plotted as a function of cross-streaam co-ordinate at (a) x/H=3, (b)

x/H=4, (c) x/H=5, (d) x/H=7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

12 Streamwise evolution of turbulent kinetic energy for (a) Ma2, (b) Ma3, (c) Ma4. . . . 47

13 Cross-stream integrated turbulent kinetic energy. . . . . . . . . . . . . . . . . . . . 47

14 Streamwise evolution of peak normal stresses. . . . . . . . . . . . . . . . . . . . . . 48

40

15 Anisotropy of normal stresses at x/H=4 for (a) Ma2, (b) Ma3 and (c) Ma4. . . . . . 48

16 Balance of turbulent kinetic energy budget for Ma4. . . . . . . . . . . . . . . . . . . 49

17 Production of turbulent kinetic energy: (a) x/H=2.5, (b) x/H=3.0, (c) x/H=3.5, (d)

x/H=4.0, (e) x/H=4.5, (f) x/H=5.0. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

18 Production of turbulent kinetic energy in the reattachment region for (a) x/H=5.5 and

(b) x/H=6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

19 Cross-stream integrated production of turbulent kinetic energy. . . . . . . . . . . . . 51

20 Streamwise evolution of (a) shear stress, (b) production for Ma2 (top) and Ma4 (bottom). 51

21 Cross-stream integrated shear stress. . . . . . . . . . . . . . . . . . . . . . . . . . . 52

22 Solenoidal dissipation at (a) x/H=2.5, (b) x/H=3, (c) x/H=3.5, (d) x/H=4.0, (e)

x/H=4.5, (f) x/H=5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

23 Dilatational dissipation at (a) x/H=2.5, (b) x/H=3, (c) x/H=3.5, (d) x/H=4.0,(e)

x/H=4.5, (f) x/H=5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

24 Pressure dilatation at (a) x/H=2.5, (b) x/H=3, (c) x/H=3.5, (d) x/H=4.0, (e) x/H=4.5,

(f) x/H=5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

25 Solenoidal dissipation (lines) and pressure dilatation (lines with symbols) in the reat-

tachment region for (a) x/H=5.5 and (b) x/H=6.5. . . . . . . . . . . . . . . . . . . 55

26 Turbulent diffusion of k at (a) x/H=3.5, (b) x/H=4.0, (c) x/H=4.5, (d) x/H=5, (e)

x/H=5.5, and (f) x/H=6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

27 Pressure diffusion of k at (a) x/H=3.5, (b) x/H=4.0, (c) x/H=4.5, (d) x/H=5, (e)

x/H=5.5, and (f) x/H=6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

41

5

6

1

X

ZY

17.92

10.76

Figure 1: Schematic of the backward-facing step configuration

0 0.5 1U

0

0.5

1

1.5

2

z/H

Exp.p=6p=8p=10

0 0.5 1U

0

0.5

1

1.5

2

0 0.5 1U

0

0.5

1

1.5

2

0 0.5 1U

0

0.5

1

1.5

2

-0.2 -0.1 0W

0

0.5

1

1.5

2

z/H

-0.2 -0.1 0W

0

0.5

1

1.5

2

-0.2 -0.1 0W

0

0.5

1

1.5

2

-0.2 -0.1 0W

0

0.5

1

1.5

2

x/H=7x/H=5x/H=4x/H=3

Figure 2: Convergence study through comparison of averaged velocities. Streamwise velocity (top),

wall normal velocity (bottom) for Ma=0.4 for different polynomial approximation orders.

42

0 0.5 1U

0

0.5

1

1.5

2

z/H

Exp.

Ma2Ma3Ma4

0 0.5 1U

0

0.5

1

1.5

2

0 0.5 1U

0

0.5

1

1.5

2

0 0.5 1U

0

0.5

1

1.5

2

-0.2 -0.1 0W

0

0.5

1

1.5

2

z/H

-0.2 -0.1 0W

0

0.5

1

1.5

2

-0.2 -0.1 0W

0

0.5

1

1.5

2

-0.1 -0.05 0W

0

0.5

1

1.5

2

x/H=3 x/H=4 x/H=5 x/H=7

Figure 3: Comparison of averaged streamwise velocity (top), and wall normal velocity (bottom) with

the experiment of incompressible flow12.

0 0.02 0.04 0.06 0.08{uu}

0

0.5

1

1.5

2

z/H

0 0.02 0.04 0.06{uu}

0

0.5

1

1.5

2

0 0.01 0.02 0.03{uu}

0

0.5

1

1.5

2

p=6p=8

0 0.01 0.02 0.03 0.04{uw}

0

0.5

1

1.5

2

z/H

0 0.01 0.02 0.03{uw}

0

0.5

1

1.5

2

0 0.01 0.02{uw}

0

0.5

1

1.5

2

x/H=4 x/H=6 x/H=8

Figure 4: Convergence study through comparison of Favre averaged stresses for Ma=0.4 for different

polynomial approximation orders.

43

0 0.04 0.08{uu}

0

0.5

1

1.5

2

z/H

Exp.Ma2Ma3Ma4

0 0.04 0.08{uu}

0

0.5

1

1.5

2

0 0.02 0.04{uu}

0

0.5

1

1.5

2

0 0.02 0.04{uw}

0

0.5

1

1.5

2

z/H

0 0.02 0.04{uw}

0

0.5

1

1.5

2

0 0.01 0.02{uw}

0

0.5

1

1.5

2

x/H=4 x/H=6 x/H=8

Figure 5: Comparison of Favre averaged streamwise normal stress (top), and shear stress (bottom)

with the experiment of incompressible flow12.

Figure 6: Pressure iso-contours indicating the flow topology for the case Ma4.

44

Figure 7: Vortex cores depicting the helical density in the flow for the case Ma4.

High speed side

Low speed side

Outflow

Mean flow direction

L x

InflowLy

x

y

Figure 8: Schematic of the free shear layer configuration.

-2 -1 0 1 2y/δ

w(x)

0

0.25

0.5

0.75

1

U-U

2

x*/d

w(0)=50, Ma=0.25

x*/d

w(0)=75, Ma=0.25

Stanley and Sarkar

Figure 9: Mean streamwise velocity in self similar co-ordinates for two-dimensional free shear layer.

45

0 100 200 300 400x

5

10

15

δ w(x

)

Ma=0.25Ma=0.5Ma=0.6

Figure 10: Growth of vorticity thickness of two-dimensional shear layer for different Mach numbers.

0.95 0.96 0.97 0.98 0.99 10

1

2

3

z/H

Ma2Ma3Ma4

0.96 0.98 10

1

2

3

0.98 0.99 1 ρ

0

1

2

3

z/H

0.99 1 1.01 1.02 ρ

0

1

2

3

(a) (b)

(c) (d)

Figure 11: Averaged density plotted as a function of cross-streaam co-ordinate at (a) x/H=3, (b)

x/H=4, (c) x/H=5, (d) x/H=7.

46

0 0.5 1 1.5 2z/H

0

0.05

0.1

k

0 0.5 1 1.5 2z/H

0

0.05

0.1

x/H=0.5x/H=1x/H=1.5x/H=2x/H=2.5x/H=3x/H=3.5x/H=4x/H=4.5x/H=5

0 0.5 1 1.5 2z/H

0

0.05

0.1

(a) (b) (c)

Figure 12: Streamwise evolution of turbulent kinetic energy for (a) Ma2, (b) Ma3, (c) Ma4.

0 1 2 3 4 5x/H

0

0.02

0.04

0.06

0.08

k (c

ross

-str

eam

inte

grat

ed)

Ma2Ma3Ma4

Figure 13: Cross-stream integrated turbulent kinetic energy.

47

2 3 4 5 6 7 80

0.04

0.08

{uu}

Ma2Ma3Ma4

2 3 4 5 6 7 80

0.04

0.08

{vv}

2 3 4 5 6 7 8x/H

0

0.04

0.08

{ww

}

Figure 14: Streamwise evolution of peak normal stresses.

0 0.5 1 1.5 2-0.4

-0.2

0

0.2

0.4

b ij

buu

bvv

bww

0 0.5 1 1.5 2-0.4

-0.2

0

0.2

0.4

b ij

0 0.5 1 1.5 2z/H

-0.4

-0.2

0

0.2

0.4

b ij

(a)

(b)

(c)

Figure 15: Anisotropy of normal stresses at x/H=4 for (a) Ma2, (b) Ma3 and (c) Ma4.

48

0.5 1 1.5 2

0

0.01

0.02

0.03

Bud

get

LHSRHS

0.5 1 1.5 2z/H

-0.02

-0.01

0

0.01

0.02

0.03

Bud

get

x/H=3.5

x/H=4

Figure 16: Balance of turbulent kinetic energy budget for Ma4.

0.5 1 1.5 20

0.02

0.04

0.06

0.08

Prod

uctio

n of

k

Ma2Ma3Ma4

0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.5 1 1.5 2z/H

0

0.02

0.04

0.06

0.08

Prod

uctio

n of

k

0.5 1 1.5 2z/H

0

0.02

0.04

0.06

0.08

0.5 1 1.5 2z/H

0

0.02

0.04

0.06

0.08

(a) (b) (c)

(d) (e) (f)

Figure 17: Production of turbulent kinetic energy: (a) x/H=2.5, (b) x/H=3.0, (c) x/H=3.5, (d)

x/H=4.0, (e) x/H=4.5, (f) x/H=5.0.

49

0.5 1 1.5 20

0.02

0.04

0.06

Prod

uctio

n of

k Ma2Ma3Ma4

0.5 1 1.5 2z/H

0

0.01

0.02

0.03

Prod

uctio

n of

k

(a)

(b)

Figure 18: Production of turbulent kinetic energy in the reattachment region for (a) x/H=5.5 and (b)

x/H=6.5.

50

0 1 2 3 4 5x/H

0

0.005

0.01

0.015

0.02

0.025

0.03

Cro

ss-s

trea

m in

tegr

ated

pro

duct

ion

of k

Ma2Ma3Ma4

Figure 19: Cross-stream integrated production of turbulent kinetic energy.

0 0.5 1 1.5 2-0.04

-0.03

-0.02

-0.01

0

{uw

}

x/H=2.5x/H=3x/H=3.5x/H=4x/H=4.5

0.5 1 1.5 20

0.02

0.04

0.06

0.08

Prod

uctio

n of

k

0 0.5 1 1.5 2z/H

-0.04

-0.03

-0.02

-0.01

0

{uw

}

0.5 1 1.5 2z/H

0

0.02

0.04

0.06

0.08

Prod

uctio

n of

k

(a) (b)

(a) (b)

Figure 20: Streamwise evolution of (a) shear stress, (b) production for Ma2 (top) and Ma4 (bottom).

51

2 2.5 3 3.5 4 4.5 5x/H

0

0.01

0.02

0.03

Cro

ss-s

trea

m in

tegr

ated

{uw

}

Ma2Ma3Ma4

Figure 21: Cross-stream integrated shear stress.

52

0.5 1 1.5 20

0.002

0.004

0.006

ε s

0.5 1 1.5 20

0.005

0.01

0.015

0.5 1 1.5 20

0.01

0.02

0.03

Ma2Ma3Ma4

0.5 1 1.5 2

z/H

0

0.01

0.02

0.03

ε s

0.5 1 1.5 2

z/H

0

0.01

0.02

0.03

0.5 1 1.5 2

z/H

0

0.01

0.02

0.03

(a) (b) (c)

(d) (e) (f)

Figure 22: Solenoidal dissipation at (a) x/H=2.5, (b) x/H=3, (c) x/H=3.5, (d) x/H=4.0, (e) x/H=4.5,

(f) x/H=5.

53

0.5 1 1.5 20

5e-06

1e-05

1.5e-05

2e-05

ε d

0.5 1 1.5 20

1e-05

2e-05

3e-05

4e-05

5e-05

0.5 1 1.5 20

0.0001

0.0002Ma2Ma3Ma4

0.5 1 1.5 2

z/H

0

0.0001

0.0002

ε d

0.5 1 1.5 2

z/H

0

0.0001

0.0002

0.5 1 1.5 2

z/H

0

0.0001

0.0002

(a) (b) (c)

(d) (e) (f)

Figure 23: Dilatational dissipation at (a) x/H=2.5, (b) x/H=3, (c) x/H=3.5, (d) x/H=4.0,(e) x/H=4.5,

(f) x/H=5.

0.5 1 1.5 2-0.002

-0.0015

-0.001

-0.0005

0

Pres

sure

-dila

tatio

n

0.5 1 1.5 2-0.004

-0.003

-0.002

-0.001

0

0.5 1 1.5 2-0.008

-0.006

-0.004

-0.002

0

Ma2Ma3Ma4

0.5 1 1.5 2

z/H

-0.02

-0.015

-0.01

-0.005

0

Pres

sure

-dila

tatio

n

0.5 1 1.5 2

z/H

-0.02

-0.015

-0.01

-0.005

0

0 0.5 1 1.5 2

z/H

-0.02

-0.015

-0.01

-0.005

0

(a) (b) (c)

(d) (e) (f)

Figure 24: Pressure dilatation at (a) x/H=2.5, (b) x/H=3, (c) x/H=3.5, (d) x/H=4.0, (e) x/H=4.5,

(f) x/H=5.

54

0.5 1 1.5 2-0.02

0

0.02

0.04

ε s, p-d

ilata

tion Ma2

Ma3Ma4

0.5 1 1.5 2z/H

-0.01

0

0.01

0.02

ε s, p-d

ilata

tion

(a)

(b)

Figure 25: Solenoidal dissipation (lines) and pressure dilatation (lines with symbols) in the reattach-

ment region for (a) x/H=5.5 and (b) x/H=6.5.

0.5 1 1.5 2-0.04

-0.02

0

0.02

0.04

Tur

bule

nt d

iffu

sion

Ma2Ma3Ma4

0.5 1 1.5 2-0.04

-0.02

0

0.02

0.04

0.5 1 1.5 2-0.04

-0.02

0

0.02

0.04

0.5 1 1.5 2

z/H

-0.04

-0.02

0

0.02

0.04

Tur

bule

nt d

iffu

sion

0.5 1 1.5 2

z/H

-0.04

-0.02

0

0.02

0.04

0.5 1 1.5 2

z/H

-0.04

-0.02

0

0.02

0.04

(a) (b) (c)

(d) (e) (f)

Figure 26: Turbulent diffusion of k at (a) x/H=3.5, (b) x/H=4.0, (c) x/H=4.5, (d) x/H=5, (e)

x/H=5.5, and (f) x/H=6.5.

55

0.5 1 1.5 2-0.04

-0.02

0

0.02

0.04

Pres

sure

-dif

fusi

on

Ma2Ma3Ma4

0.5 1 1.5 2-0.04

-0.02

0

0.02

0.04

0.5 1 1.5 2-0.04

-0.02

0

0.02

0.04

0.5 1 1.5 2

z/H

-0.04

-0.02

0

0.02

0.04

Pres

sure

-dif

fusi

on

0.5 1 1.5 2

z/H

-0.04

-0.02

0

0.02

0.04

0.5 1 1.5 2

z/H

-0.04

-0.02

0

0.02

0.04

(a) (b) (c)

(d) (e) (f)

Figure 27: Pressure diffusion of k at (a) x/H=3.5, (b) x/H=4.0, (c) x/H=4.5, (d) x/H=5, (e) x/H=5.5,

and (f) x/H=6.5.

56