computational fluid complex flows, multiscale physics...

Computational Fluid Dynamics


http://www.nd.edu/~gtryggva/CFD-Course/

Grétar Tryggvason

Lecture 24April 24, 2017


Complex Flows,Multiscale Physics

&Direct Numerical

Simulations



For the vast majority of problems of practical interest, we must deal with many physical processes taking place over a range of spatial and temporal scales.

For the most part, those are dealt with by treating unresolved processes using models.

In some cases the models are so well established and we can apply them with such confidence that we take the models to to be an “exact” description of the physics.

This is the case for continuum descriptions of many materials


Bridging the range of scales between atomistic description of materials and processes and continuum scale is a large and active research area, but not the subject of the present course.

For many flow systems we have good reasons to believe that the continuum description is an accurate one, but the range of continuum scales is such that a fully resolved direct solution of the continuum equations for industrial systems is not practical.


Some of the best known examples include:

Turbulent flows

Multiphase flows (both fluid-fluid and fluid-solid)

Reacting flows

And, of course

Reacting, multicpase turbulent flows!


Single PhaseTurbulent Flow



Examples for turbulent flow

https://www.youtube.com/watch?v=HDY-szDEH9s

https://www.youtube.com/watch?v=WSnTfP_U3yE

https://www.youtube.com/watch?v=aXysnLWlWZg

https://www.youtube.com/watch?v=wXsl4eyupUY&t=65s

https://www.youtube.com/watch?v=10ZCn6KCRYs


For single phase flow under “normal” conditions we have every reason to believe that the Navier-Stokes equations fully describe the flow.

Thus, solving the governing equations for unsteady flow where all continuum temporal and spatial scales are fully resolved should reproduce the fluid motion. If, in particular, the flow is unsteady then the solution should show how the instability grows and the motion becomes turbulent.


Examples of Reynolds numbers:Flow around a 3 m long car at 100 km/hr:

Flow around a 100 m long submarine at 10 km/hr:

�

Re = LUv

= 3× 27.781.5 ×10−5

= 5.5 ×106

Kinematic viscosity (~20 °C)

Water ν = 10-6 m2/sAir ν = 1.5 ✕10-5 m2/s

1km/hr = 0.27778 m/s

�

Re = LUv

= 100 × 2.7810−6

= 2.78 ×108

Water flowing though a 0.01 m diameter pipe with a velocity of 1 m/s

�

Re = LUv

= 0.01×110−6

=104


It can be shown that for turbulent flow the ratio of the size of the smallest eddy to the length scale of the problem

�

δL≈O(Re−3 / 4 )

If about 10 grid points are needed for Re=10 (the driven cavity problem)

Re 3d 2d103~ 3003 ~ 1002

104 ~ 20003 ~ 3002

105~ 100003 ~ 10002

Largest computations today use about 100003 points

�

δL≈O(Re−1/ 2)

In 3D In 2D


Computational Modeling of Turbulent Flow:Direct Numerical Simulations (DNS): Every length and time scale is fully resolvedReynolds Averaged Navier-Stokes (RANS): Only the averaged motion is computed. The effect of fluctuations is modeled

Large Eddy Simulations (LES): Large scale motion is fully resolved but small scale motion is modeled


The use of DNS results to assess the quality of subgrid models can be done in two ways:

A priori tests: Averaging the DNS results in the appropriate ways and comparing with the model predictions.

A posteriori: Implementing the model into an reduced order simulation and comparing the predictions with DNS results, averaged in the appropriate way.


Reynolds Averaged

Navier-Stokes Equations


To solve for the mean motion, we derive equations for the mean motion by averaging the Navier-Stokes equations. The velocities and other quantities are decomposed into the average and the fluctuation part

a = A + a'

< a > = A< a' > = 0< a + b > = A+ B< ca > = cA< ∇a > =∇A

Defining an averaging procedure that satisfies the following rules:

This will hold for spatial averaging, temporal averaging, and ensamble averaging


�

∂∂tu + ∇ ⋅uu = − 1

ρ∇p+ ν∇2u

u =U + u'p = P+ p'

< a >=A< a' >= 0< ca >=cA< ∇a >=∇A

Start with the Navier-Stokes equations

Decompose the pressure and velocity into mean and fluctuations:

a = A + a'

Or, in general, for any dependant variable:


∂∂tU +∇ ⋅UU = −

1ρ∇P + ν∇2U − ∇⋅ < u'u' >

Applying the averaging to the Navier-Stokes equations results in:

�

< u'u'>=< u'u'> < u'v'> < u'w'>< u'v'> < v 'v'> < v 'w'>

< u'w'> < v 'w'> < w'w'>

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Reynold’s stress tensor

Since we only have an equation for the mean flow, the Reynolds stresses must be related to the mean flow through closure relations.


Zero and One equation models


Introduce the “turbulent eddy viscosity”

�

νT = l02

t0

< u'u' >ij= −νT

∂Ui

∂x j

+∂U j

∂xi

⎛

⎝⎜

⎞

⎠⎟

where


Zero equation models

�

νT = l02 dUdy

Prandtl’ mixing length

�

l0 = κy

Smagorinsky model

Baldvin-Lomaz model

�

νT = l02 2S ijS ij( )1/ 2

�

νT = l02 ω iω i( )1/ 2

�

S ij = 12

∂Ui

∂x j

+∂U j

∂xi

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

�

ω i = ∂Ui

∂x j

−∂U j

∂xi

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


One equation models

�

νT = k1/ 2t0

Where k is obtained by an equation describing its temporal-spatial evolution

However, the problem with zero and one equation models is that t0 and l0 are not universal. Generally, it is found that a two equation model is the minimum needed for a proper description


Two equation models


To characterize the turbulence it seems reasonable to start with a measure of the magnitude of the velocity fluctuations. If the turbulence is isotropic, the turbulent kinetic energy can be used:

�

k = 12

< u'u'> + < v 'v'> + < w'w'>( )The turbulent kinetic energy does, however, not distinguish between large and small eddies.


To distinguish between large and small eddies we need to introduce a new quantity that describe

�

ε ≡ ν ∂u'i∂u'i∂x j∂x j

Usually, the turbulent dissipation rate is used

Smaller eddies dissipate fasterversus

�

νT = Cµk 2

ε�

∂∂tU + ∇ ⋅UU = − 1

ρ∇P + ν + νT( )∇2U

Solve for the average velocity

Where the turbulent kinematic eddy viscosity is given by


�

∂k∂t

+U j∂k∂x j

= τ ij∂Ui

∂x j

−ε + ∂∂x j

ν ∂k∂x j

− 12ui'ui'u j' − 1

ρp'u j

'⎡

⎣ ⎢

⎤

⎦ ⎥

The exact k-equation is:

where

�

τ ij = − ui'u j'

The exact epsilon-equation is considerably more complex and we will not write it down here.

Both equations contain transport, dissipation and production terms that must be modeled


∂k∂t

+U ⋅∇k = ∇ ⋅Dk∇k + production − dissipation

∂ε∂t

+U ⋅∇ε = ∇ ⋅Dε∇ε + production − dissipation

The general for for the equations for k and epsilon is:

These terms must be modeled Closure involves proposing a form for the missing terms and optimizing free coefficients to fit experimental data


Here

The k-epsilon model

�

νT = C k 2

ε

�

τ ij =< ui'u j' >= 2

3kδij −νT

∂Ui

∂x j

+∂U j

∂xi

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ and

�

C1 = 0.09; C2 =1.0; C3 = 0.769; C4 =1.44; C5 =1.92

�

DkDt= +∇ ⋅ (ν + C2νT )∇k - τ ij

∂Ui

∂x j

−ε

�

DεDt= ∇ ⋅ (ν + C3νT )∇ε + C4

εkτ ij

∂Ui

∂x j

−C5ε2

kProduction Dissipation

Turbulenttransport


Two major numerical difficulties

The equations may be stiff in some regions of the flow requiring very small time step. This can be overcome by an implicit scheme.

In reality k goes to zero at the walls. In simulations this usually takes place so close to the wall that it is not resolved by the grid. To overcome this we usually use a “wall function” or a damping function. See: Patel, Rodi, and Scheuerer, “Turbulence Models for Near-Wall and Low Reynolds Number Flows: A Review. AIAA Journal, 23 (1985), 1308-1319


Other two equation turbulence models:RNG k-epsilonNonlinear k-epsilonk-enstrophyk-lok-reciprocal timeetc


Second order closure


The k-epsilon and other two equation models have several serious limitations, including the inability to predict anisotropic Reynolds stress tensors, relaxation effects, and nonlocal effects due to turbulent diffusion.

For these problems it is necessary to model the evolution of the full Reynolds stress tensor


Derive equations for the Reynolds stresses:

∂ui

∂t+∇uiu j = −

1ρ∇p + ν∇2ui

The Navier-Stokes equations in component form:

�

ui∂ui∂t

+ ∇uiu j = - 1ρ ∇p+ ν∇2ui⎛ ⎝ ⎜

⎞ ⎠ ⎟

Multiply the equation by the velocity

and averaging leds to equations for

�

∂∂t

uiu j


The new equations contain terms like

which are not known. These terms are therefore modeled

�

uiuiu j

The Reynolds stress model introduces 6 new equations (instead of 2 for the k-e model. Although the models have considerably more physics build in and allow, for example, anisotrophy in the Reynolds stress tensor, these model have yet to be optimized to the point that they consistently give superior results.

For practical problems, the k-e model or more recent improvements such as RNG are therefore most commonly used!


Turbulence models are used to allow us to simulate only the averaged motion, not the unsteady small scale motion.

Turbulence modeling rest on the assumption that the small scale motion is “universal” and can be described in terms of the large scale motion.

Although considerable progress has been made, much is still not known and results from calculations using such models have to be interpreted by care!


For more information:D. C. Wilcox, Turbulence Modeling for CFD (2nd ed. 1998; 3rd ed. 2006). The author is one of the inventors of the k-ω model and the book promotes it use. The discussion is, however, general and very accessible, as well as focused on the use of turbulence modeling for practical applications in CFD. However, the focus is mostly on relatively classical ideas.


Newer work tends to:•  Accept that it is unlikely that completely

steady-state models will capture all scales•  Recognize the severe limitation that the

eddy viscosity concept places on the complexity of processes that can be captured

•  Direct Numerical Simulations are providing enormous input into the nature of the closure terms


Large Eddy Simulations


Turbulent flow generally contains a range of fluid structures of sizes ranging from the domain to the dissipation scale. Thus, there is no real separation of scales and RANS averaging attempts to capture the complete range.

In Large Eddy Simulations (LES) the large scale motion is resolved but the smallest scales modeled.

In its original form only the very smallest scales were modeled but current practice includes various cutoffs, sometimes referred to as Very Large Eddy Simulations (VLES) or Unsteady RANS (URANS).


For LES the Navier Stokes equations are filtered

The sub-grid scale (SGS) models

∂ui∂t

+∇uiu j = −1ρ∇p+ν∇2ui −∇τ ij

τ ij = uiu j −uiu j


Simplest: Smagorinsky model

τ ijSmag = −2ν sgSij = −2 csΔ( ) S Sij

Sij =12∂ui∂x j

+∂u j∂xi

⎛

⎝⎜⎜

⎞

⎠⎟⎟

where

cs ≈ 0.16; Δ


More complex models

Evolution equation for SGS kinetic energy which is then used to compute the eddy viscosity

Dynamic models, where the flow field is filtered at a larger scale and the results used to extrapolate to the unresolved scales (Germano identity)

These models do not include the “backscatter” where kinetic energy is transferred from the unresolved scales to the resolved ones

Special wall-functions are typically used near walls


http://www.scholarpedia.org/article/Turbulence:_Subgrid-Scale_Modeling


Multiphase Flow


Need model equations to predict flow rates, pressure drop, slip velocities, and void fraction

Mixture models: one averaged phase

Two-fluid models: two interpenetrating continuum


This figure shows schematically one of several different configurations of a circulating fluidized bed loop used in engineering practice. The particles flow downward through the aerated “standpipe”, and enter the bottom of a fast fluidized bed “riser”. The particles are centrifugally separated from the gas in a train of “cyclones”. In this diagram, the particles separated in the primary cyclone are returned to the standpipe while the fate of the particles removed from the secondary cyclone is not shown.

From: Computational Methods for Multiphase Flow, Edited by A.Prosperetti and G.Tryggvason


Although commercial codes will let you model relatively complex multiphase flows, it is really only in the limit of dispersed and dilute flows where we can expect reasonable accuracy

To treat systems like this, the two-fluid model is usually used. The continuous phase is almost always used in an Eularian way where the continuity, momentum, and energy equations are solved on a fixed grid.


The void fraction εp describes how much of the region is occupied by phase p. Obviously:

�

εp =1∑While the averaging is similar to turbulent flows, here we must account for the different phases

�

α p =1 inside phase p0 otherwise

⎧ ⎨ ⎩

The void fraction is found by

�

εp = 1V

α p dvV∫

The effective density of phase p is

�

ˆ ρ = εpρp


�

ˆ φ p = 1εpV

α pφ dvV∫Averages are found by

Where the volume V goes to zero in some way

The velocity is found by

�

ˆ u p = 1εpV

upα p dvV∫The averages can also be interpreted as time or ensemble averages

The effective density of phase p is

�

ˆ ρ = εpρp

�

α pρp dvV∫

The total mass of phase p in a control volume is


The conservation of momentum equation becomes

�

∂∂t

εpρpup( ) + ∇ ⋅ εpρpupup( ) = −εp∇pp

+∇ ⋅ εpµpDp( ) + εpρpg + ∇ ⋅ εpρp < uu >( ) + Fint

In addition to the Reynolds stresses, it is now necessary to model the interfacial forces. The kinetic energy is often neglected, even though the fluctuations are non-zero in laminar flow

interfacial forces

Reynolds stresses

�

∂∂tεpρp + ∇ ⋅ εpρpup( ) = ˙ m p

The mass conservation equation can be averaged to yield

Here

�

˙ m p = 0∑


Euler/Euler approachAll phases are treated as interpenetrating continuumThe dispersed phase is averaged over each control volumeEach phase is governed by similar conservation equationsModeling is needed for

interaction between the phasesturbulent dispersion of particlescollision of particles with walls

A size distribution requires the solution of several sets of conservation equations Numerical diffusion at phase boundaries may result in errorsThis approach is best suited for high volume fraction of the dispersed phase


Euler/Lagrange approachThe fluid flow is found by solving the Reynolds-averaged Navier-Stokes equations with a turbulence model.The dispersed phase is simulated by tracking a large number of representative particles.A statistically reliable average behavior of the dispersed phase requires a large number of particlesThe point particles must be much smaller than the grid spacingModeling is needed for

collision of particles with wallsparticle/particle collisions and agglomerationdroplet/bubble coalescence and breakup

A high particle concentration may cause convergence problems


If there is no mass transfer m=0 and F is the force that one phase exerts on the other

�

Fp = 0∑In principle the conservation equations can be solved for both the continuous and the dispersed phase (Euler/Euler approach).

However, the dispersed phase is not all that continuous and an other approach is to explicitly tract (representative) particles by solving

�

dudt

= Fp

If the particles have no influence on the fluid: One way couplingIf the particles exert a force on the fluid: Two way coupling


where kD =

34

CDεrρq

ur − up( )dr

�

CD = CD Re( )

is obtained from experimental correlations, such as

and

�

CD = 24Re

1+ 0.15Re0.687( ) Re <103

For solid particles Re based on slip velocity

�

Fp = kD u−up( ) + g ρD − ρρ

+ Fother

Usually the force is written:

Other forces due to added mass, pressure, lift, etcGravity

buoyancyDrag force


For turbulent flow, set particle velocity

�

up + u'

Random velocity fluctuations from

This allows particles to cross streamlines as they do in turbulent flow

Particles can accumulate here

�

kp = u'u'∑

The force allows us to find the particle velocity by integrating:

�

dupdt

= Fp and trajectories by

�

dx pdt

= up

Usually a large number of particles is used to get a well converged particle distribution

Notice that almost all the interactions (particles/flow) particle/particle, particle/wall) are highly empirical


Similar approach can be taken for the temperature and the size of a particle (heat and mass transfer)

�

mpcpdTpdt

= hAp (Tf −Tp ) + εpApσ (T∞4 −Tp

4 )

�

dmp

dt= ˙ m p

For dilute flows this does work reasonably well — if the initial or inlet conditions are knows

Mass transfer due to evaporation, for example


Turbulent in the continuous phase

�

DkDt

=+ <U ⋅Fp >

�

<U ⋅Fp >= τρ

< uf (uf − up ) >= τρ(< uf uf > − < uf up >)

This term can lead to both reduction and increase in the turbulence in the liquid

Either ignore the contributions of the dispersed phase when computing the flow, or use a k-ε modelSolve for k and ε in the liquid and kp. Called k ε kp models. The k equation is


The full two-fluid model suffers from several problems, in addition to uncertainties about the various closure assumptions:

The major one is that the full equations are ill-posed and one cannot expect a fully converged solution under grid refinement

One possible way around this is to use the “drift flux approximation” where the particle velocity is assumed to be a given function of the local conditions.


For more information about computing multiphase flow, see:

Modeling of multiphase flows is still a very immature area. Interpret the results with care!


UsingDirect Numerical

Simulations (DNS) to Model

Turbulent Flows


DNS results contain detailed information about the flow at every spatial and temporal location so any statistic and closure terms can be computed exactly

Data set obviously must contain sufficiently large range of temporal and spatial scales

Generally the domain must be sufficiently large so that the turbulent flow is sustained and the larger than any correlation length


Simulations of turbulent channel flows, with periodic boundaries, are perhaps the most common setup, but other flows have also been simulated.

Generally the results are in complete agreement with experimental measurements

In addition to computing model terms directly, DNS results have been used extensively in various other ways, including to identify structures and find non-traditional reduced order models


Proper Orthogonal Decomposition


u ' x, t( ) = ai (t)i=1

N

∑ vi x( )

Decompose the flow field using optimum basis functions

u x, t( ) = u0 x( )+u ' x, t( )

For unit basis vectors such that

viTvi =1

We want to select the basis vectors such that each on carries the maximum energy


Construct a dataset consisting of n velocities at m times

X =X11 ! X1

m

" # "Xn1 ! Xn

m

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

The projection of the data on a basis vector i is given by viTX

And we want the variance

to be maximum, subject to the condition that the basis vectors are unit vectors

Var viTX⎡⎣ ⎤⎦= vi

TXXTvi


To maximize we use

giving

L vi,λ( ) = viTXXTvi −λ viTvi −1( )

∂L∂vi

= 2XXTvi − 2λvi = 0

∂L∂λ

= − viTvi −1( ) = 0

XXTvi = λvi

Mvi = λviM =XXT

or

Introducing We have


A singular value decomposition is X =UΣVT

m

n nr r

UVT rΣ

X

Where the diagonal terms are the eigenvalue of

Mvi = λvi M =XXTwhere


The modes are given by U

The amplitudes are given by a = ΣVT

A reduced order solution is obtained by keeping only the first few eigenvalues

EXAMPLE

And the solution is at time m is given by Xim

Computational Fluid Dynamics Computational Fluid Dynamics


Machine Learning to Classify and

Correlate



Summary


One might be tempted to think that such a well studied problem as single phase turbulence would be solved by now. That is, of course, not the case. We can, however, expect significant progress

Computers are now large enough to allow simulations with sufficiently large range of scales so that it is not clear that increasing it further will make much of a difference

New ideas are emerging what to do with the results and how to use them to generate reduced order models

It is likely that the emphasize will rapidly turn to more complex problems, such as multiphase and reacting flows

computational fluid complex flows, multiscale physics...

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