10 simulations of turbulence(cancelled)

8/10/2019 10 Simulations of Turbulence(Cancelled)

1/53

Turbulent Flows Simulations

1. Theory of Turbulent Flows

Highly unsteady, random

3 D

A great deal of voriticity

Turbulent diffusion

Coherent (organized) structures

Fluctuate on a broad range of length and time scales

Experimental Observations


2/53

Mixing layer, Brown and Roshko(1956)

Cantwell(1981)

Aluminum flakes

suspended in water

Re=4300


3/53

Falco(1977)Turbulent boundary layer Re=4000

A fog of tiny oil droplets + sheet of light


4/53

Kline(1967)

Low- and high-speed streaks at y+=2.7


5/53

Robinson(1990)

Near-wall region

Outer region

Wallace(1972)


6/53

Ejection / Sweep / BurstHinze(1975)

Willmarth(1972)

during bursting60 , 30.5u v u v y

Friction


7/53

Iso-surfaces of the second invariant of the velocity gradient tensor

Flat plate boundary layer (DNS) Wu and Moin(2009)


8/53

Coherent Structures

Practical engineering method

Truly prediction theory

slow

Not yet


9/53

Energy cascade(Richardson,1922)

Energy is transferred to successively smaller and smaller eddies

until the smallest eddies where molecular viscosity is effective

in dissipating the kinetic energy.

Kolmogorov hypotheses(1941)

I. Kolmogorovs local isotropy hypothesis:

At sufficient high Re, the small scale turbulent motions (l


10/53

II. Kolmogorovs first similarity hypothesis:

At sufficient high Re, the statistics of the small scale turbulent motions (l


11/53

III. Kolmogorovs second similarity hypothesis:

At sufficient high Re, the statistics of the small scale turbulent motions

(


12/53

The energy spectrum

How the turbulent kinetic energy is distributed among the eddies of

different sizes.

( )k E d turbulent kinetic energy Wave number

Turbulent energy spectrum 2/3 5/3( )E C C is constant.

With dimensional analysis and the Kolmogorovs local isotropic

and similarity hypotheses


13/53

(Gotoh,2002))

Isotropic turbulence,Symbol: DNS


14/53

Energy spectrum of turbulence

in function of wave number k, with

indication of the range of application

of the DNS, LES and RANS models.

The Taylor length scales lTand integral scale lI are

associated with the LES and

RANS approximations,

respectively

Universal equilibrium range


15/53

The biggest progress in turbulence

research is in turbulence modeling. CFD

software industry is based on turbulence

modeling. Many people now make their

livings on CFD softwares.

Dr.T.Gatski,

2. Turbulence modelling


16/53

History of Turbulence modelling

Time-averaged(1889)

Mixing-length model(1925)

Log law(1930)

Second moment

Closures(1945)

K-model(1972) Second moment

Closures-LRR(1975)LES(1963)

SA-DES(1997)

Two-Eqs. RANS-LES(2001)


17/53

Turbulent flows are governed by Navier-Stokes equations.

DNS (Direct Numerical Simulation )

Largest scale

integral scaleL

Smallest scale

3 / 4ReL

L

ReLis based on the magnitude of the velocity fluctuation and the integral scale.

ReL0.01 Re

How to solve NS directly?

(1) Resolution

Kolmogorov scale

on which viscosity is active, i.e., kinetic energy dissipation occurs

the distance over which the fluctuating component of the

velocity remains correlated.

largest turbulent eddy

Re is Reynolds number in practice.


18/53

The number of grid points

Total operations3ReL

DNS achieved now: simple flow (homogenous turbulent flow, channel flow,

free shear flow at low Re=104~105)

Applicable size: 5123 (108) grid points

(2) Numerical Methods

Explicit time advance methods----------accurate time history

2nd--4th order Runge-Kutta method

Implicitly treated for viscous terms----------numerical instability

Nondimensional sizes of the space and time3/ 4Re

L

To obtain enough information in time consequence,3 / 4ReL

the time steps needed is

3 9/ 4( ) ReLL


19/53

Accuracy is difficult to measure in DNS and LES.A small change in the initial state is amplified exponentially in time.

For equi-spaced grid and simple geometry, spectral methods is used.

Fast Fourier transform algorithm

Another difficulty is the treatment of initial and boundary conditions.

Initial BCs is obtained from the close similar simulations

On solid wall, very fine grids resolve streaksSymmetry BCs are not applicable instantaneous flow

A simulation must be run for some time before the flow develop all of the

correct characteristics of the flow------monitor some quantity

Spatial differecing-------energy conservative and low dissipation

compared to physical viscous (central scheme)

Step sizes of the space and time need to be related and the errors

in spatial and temporal discretizations should be balanced.


20/53

The results of a DNS contain very detailed information about a flow.

understand the physics

construct model

(3) The role of DNS: research tool

3. LES(Large Eddy Simulation )

Observation: large structures most energy transport

small structures little energy transport

more universal

idea: large structures compute

small structures model: sub gr id-scale model


21/53

LES : resolve large structures

DNS: resolve all kinds of structures

RANS: resolve mean flow

RANS


22/53

(1) Resolution

The number of grid points0.4ReL

Outer layer

Viscous sublayer 1.8ReL

can be appl ied at Re at least on e order of m agnitud e higher

High grid resolution is also required.

To further reduce the number of grid points, approximate wall modelis used.

(2) Numerical methodsis almost same as in DNS

more accurate than RANS and less expensive as DNS


23/53

The filtered variable at the location0r

Grepresents the f i l ter funct io n

r is the position vector

(3) Spatial filtering

Decompose

Filtered part

or resolved partSub-filter part

or unresolved part

When the sizes of the turbulent structures are less than,they are cut off.


24/53

The mostly used filter functions:


25/53

Notice that filter functions in physical space limit

both spatial and temporal resolution.


26/53

(4) Filtered governing equations

The filtered incompressible Navier-Stokes equations-----Newtonian fluid

Subg r id-Scale Stress(SGS)tensor

Describe the spatial and temporal evolution of the

large, energy-carrying scales of motion.

Describe the effects of the unresolved scales.

The SGS tensor has to be modeled to close the equations.

Subg r id-Scale Reynold s-stress tenso r

-----interactions between the small-scale structures


27/53

The filtered compressible Navier-Stokes equations

Favre(1965) averagingtogether with the spatial filtering.

Favre-averaged subg r id-scale stress tensor


28/53

(5) Subgrid-Scale modeling

Subgrid-Scale model is used to simulate energy transfer between the large

and the subgrid scales.

Transport from the large to the small ones---------cascade

small to the large ones--------backscatter

Eddy-viscosity model

incompressible

compressible

Very easy to implement in existing code

Kinetic energy dissipation is always positive----robust


29/53

Smagorinsky SGS model

Based on equilibrium hypothesis that small scales dissipate entirely

and instantaneously all the energy they received from large scales.

is modified as

Eddy viscosity

In order to account for the reduced growth of the small scales near the wall

numerically cheap and easy to implementtoo dissipative in laminar regions with mean shear

require special provisions near wall and at laminar-turbulence transition

backscatter is not modeled.

Smagorinsky constant


30/53

Dynamic SGS model

Smagorinsky constantis replaced by a parameter,

which evolves dynamically in space and time.

Based on scale similarity, i.e., the smallest resolved scale motions

can provide information that can be used for largest subgrid scale

motions (Germano et al. 1990).

automatically decrease the parameter near the wall

automatically change the parameter from much smaller in shear flows

to larger in isotropic turbulence


31/53

(6) Wall models

The costs of LES for wall-bounded flows at high Re (>106) are still too

high for engineering purposes.

Excessively large number of grid points required to resolve the wall layer

The idea: model the wall layer by specifying a correlation between the

velocity in outer flow and the stress at the wall

Basic assumption: weak interaction between the near wall and outer region.

A new zonal approach proposed by Balaras et al.(1996)

Allows it to place the first point in a region


32/53

Remarks:

At present, reasonable subgrid-scale models exist and produce

good simulations, however, the models are not sufficiently precise to

be trusted to simulate a flow that has never been treated before.

To reduce the cost of LES, with retained accuracy, reliability and

versatility, hybrid LES/RANS approaches, have recently been

developed in which RANS and LES are combined to make themost of both techniques. The most well-known model of this type

is the detached eddy simulation (DES)model, combining RANS

modeling for the attached eddies with LES computations for the

detached eddies.

Towards the use of large eddy simulation in engineering

Progress in Aerospace Sciences 44 (2008) 381396

C. Fureby

Further Reading:


33/53

4. RANS(Reynolds-Averaged Navier-Stokes equations )

------ turbulence models

(1) Basic equations of turbulence

Viscous stress tensor

Strain-rate tensor

Total energy Total enthalpy


34/53

Reyno lds averaging

Incompressible flow

mean value turbulent fluctuations

There are three different forms of Reynolds averaging

Time averaging-------statistically steady turbulence

-------incompressible flow


35/53

spatial averaging-------homogeneous turbulence


36/53

ensemble averaging-------general turbulence

In cases where the turbulence flow is both stationary and homogeneous,

all three averaging forms are equivalent-------ergodic hypothesis

Favre (mass) averagin g -------compressible flow

The most convenient way,

density and pressure--------Reynolds averaging

velocity, internal energy, enthalpy, temperature--------Favre averaging

Favre averagin g:


37/53

Favre decomposition

mean value turbulent fluctuations

Reynolds-Averaged Navier-Stokes equations

-------incompressible flow

Reynolds -stress tensor


38/53

The sum of the normal Reynolds-stresses divided by density is called

Turbulent kinet ic energy

Favre- and Reynolds-Averaged Navier-Stokes equations

------compressible flow

Favre-averaged Reynold s-stress tensor


39/53

In turbulence modeling, Morkovins hypothesis

Turbulent structure of a boundary layer is not notably influenced by

the density fluctuations if This is true generally for Ma


40/53

The proportionality factor is eddy viscosity

eddy viscosity

In compressible flow

KEY: how to model


41/53

Bouss inesq h ypothes is----------first order closures

Nonl inear Eddy-v iscosi ty

flow with sudden change of the mean strain rateflow with significant streamline curvature

flow with rotation and stratfication

flow with boundary layer separation and reattachment

secondary flow in ducts and in turbomachinery

Limitations:

Lumley proposed to extend the linear Boussinesq approach by high-order

products of strain and rotation tensor like a Taylor series expansion.

Computation work only slightly increased, but can offer a substantially

improved prediction capabilities for complex flows

----------first order closures

d d l


42/53

Reynolds -Stress transpo rt equat ion

It is possible to derive the exact equation for Reynolds-stresses

Taking time average

For incompressible flow

New unknowns

higher-order

correlations

---------second order closures

(2) Fi t d l


43/53

(2) First order closures

Boussinesq hypothesis or nonlinear eddy-viscosity

Zero-equat ion models

Turbulent fluctuations

Algebraic relations

Mean flow quantities

Underlying assumption: local rate of production and dissipation of

turbulence are approximately equal (equilibrium), and do not include

the convection of turbulence (no history effect)

Equilibriumspecify the length and velocity scale in terms of mean flow

Baldwin-Lomax (1978)

Cebeci-Smith (1974)Johnson-King (1984) ODE adverse pressure gradient

Wilcox (1988)Half-equation models


44/53

One-equat ion models

Simple and easy to incorporate into a numerical code

Flow with no separation, good results for pressure distribution

Not accurate for friction drag and rate of heat transfer.

length scale is specified algebraically

velocity scale is specified using a partial differential equation

Baldwin-Barth

Spalart-Allmaras(1994)

No need fine grid near the wall

local easy extend to unstructured grid

namely, one transport equation is derived based on the NS equations.


45/53

Two -equation mo dels

Two transport equation is derived based on the NS equations.

K-model

K-model

K kinetic energy of turbulencedissipation of turbulence

specific turbulence dissipation rate

zero equation model (Baldwin-Lomax)

one equation model (Spalart-Allmaras)two equation model (K-, K-)

Attached boundary layer: K-< BLSA < K-

Free shear layer: K-< K-


46/53

Example: An asymmetric plane diffuser, known as the OBI diffuser.

Effect of different turbulence model on the length of recirculation zone.


47/53

Pressure distribution at the bottom wall


48/53

Wall shear stress at the bottom wall


49/53

Velocity profile at one position x/H=-5.87


50/53

Velocity profile at one position x/H=19.53


51/53



52/53



53/53

The last figure is noteworthy, as it demonstrates a clear weakness of

all the tested turbulence models, in that the velocity profile in the

downstream duct of the diffuser is experimentally fully recovered,

while the calculated profiles still show remaining effects of their

earlier separation.

We can hope that the gained knowledge

on turbulence from advanced DNS and

LES simulations will contribute to the

improvement of current turbulence models.

10 simulations of turbulence(cancelled)

Documents