Introduction to Turbulence and

Turbulence Modeling !

Part I

Venkat Raman !

The University of Texas at Austin !

Lecture notes based on the book “Turbulent Flows” by S. B. Pope

Turbulent Flows

• Turbulent flows ➡ Commonly found in nature

➡ Relevant to industrial devices

➡ Contain random fluid motion

www.rug.nl/gvc/gallery/dns-zoom.jpgwww.aip.org/dbis/stories/2004/images/14254-1.jpg

http://library.thinkquest.org/03oct/01424/hurricansep_files/image002.jpg

Importance of Turbulent Flows

• Naturally occurring in practical flows ➡ Necessary for many applications

- Enhanced fuel/air mixing in internal combustion and aircraft engines

‣ Reduces size of engines drastically

➡ May not be desired in others

- Increases drag on surfaces

‣ Reduced efficiency

• Key theoretical issues ➡ How do we characterize/study turbulence?

➡ What are the essential features of turbulent flow?

Equations of Motion

• Fluid motion governed by Navier-Stokes equations

• Solved along with continuity equation

• Need to solve energy equation (if needed) along with ideal gas-law to close the system

• Density assumed to be constant here

∂ρUi

∂t+

∂ρUjUi

∂xj

= −

∂p

∂xi

+∂τij

∂xj

,

τij = µ

!

∂Ui

∂xj

+∂Uj

∂xi

"

−

2

3µδij

∂Uj

∂xj

∂ρ

∂t+

∂ρUj

∂xj

= 0

Reynolds Number

• Reynolds number is a key non-dimensional quantity

!

!

➡ Ratio of inertial to viscous forces

- Alternate definition based on length scales (to be discussed)

• High Re associated with turbulence ➡ Exact number at which transition occurs is dependent on flow

conditions

- Possible to maintain laminar flow at very high Re numbers as well

Re =⇢UL

µ=

⇢U(U/L)

µU/L2

Scales of Turbulence

• Turbulent flows involve a range of length-scales ➡ Characteristic lengths that are associated with certain flow

features

➡ A wide range of length-scales possible

One representative length-scale

Scales of Turbulence

• Turbulent motion composed of eddies of different sizes

• Let eddy of size have a velocity ➡ Associated timescale:

• Let the largest eddies be characterized by lengthscale ➡ Comparable to the flow scale

➡ Velocity of these eddies proportional to turbulence intensity

➡ Reynolds number for this size:

• What happens to these large eddies?

l u(l)

τ(l) = l/u(l)

l0

u′≡ (2/3k)1/2

Re0 ≡ u0l0/ν = u′l0/ν

L

Energy Cascade

• Large eddies are unstable and break up into smaller eddies ➡ Also transfer their energy to the smaller eddies

➡ When Reynolds number is high, the viscous forces are very small compared to the inertial forces

- Viscous dissipation is small at large eddy sizes

• Energy cascade ➡Eddies break up until the characteristic size and velocity

decrease

- Note:

- Viscous dissipation finally kills the turbulent kinetic energy

Re(l) = u(l)l/ν

Energy Dissipation Rate

• Large-scale eddies have energy ➡ Dissipation rate

• Large-scales determine the dissipation rate ➡ Break up of these eddies start the cascade process

➡ Dissipation rate is independent of viscosity

- However, viscosity itself is the cause of dissipation

➡ Verified from experiments

➡ Valid only for high Reynolds number flows

u2

0

ϵ = u2

0/(l0/u0) = u3

0/l0

Kolmogorov Hypotheses

• Hypothesis of local isotropy ➡ At sufficiently high Reynolds number, the small-scale

turbulent motions ( ) are statistically isotropic

- Implies that all geometric information is lost as energy is transferred through the length-scales

- All information about the boundaries and geometry lost

- Consider length scale defined such that

‣ Anisotropic for

‣ Isotropic for

‣ Roughly,

l << l0

lEI

l > lEI

l < lEI

lEI ≈ l0/6

Kolmogorov Hypotheses

• First similarity hypothesis ➡ In every turbulent flow at sufficiently high Reynolds

number, the statistics of small-scale motions have a universal form that is uniquely determined by and

- Leads to the following length, velocity, time scales

‣ Kolmogorov length

‣ Velocity

‣ Timescale

- Velocity and length scales decrease with eddy size

- Kolmogorov length is comparable to the size of the smallest eddies that are found in a turbulent flow

ν ϵ

η ≡

!

ν3/ϵ"1/4

uη ≡ (ϵν)1/4

τη ≡ (ν/ϵ)1/2

Reynolds Number

• Reynolds number also represents information about the range of scales

!

!

➡ As Re increases, the range of scales increases

!

!

➡ Velocity-scale at smallest scales decreases w.r.t large scale velocity

!

➡ Time-scale at the small scales also decrease w.r.t large scale

⌘

l0⇠ Re�3/4

u⌘

u0⇠ Re�1/4

⌧⌘⌧0

⇠ Re�1/2

Kolmogorov Hypotheses

• Second similarity hypothesis ➡ At sufficiently high Reynolds numbers, the statistics of the

motions of scale in the range have a universal form that is uniquely determined by , independent of

- This universal range referred to as the inertial subrange

- Motions determined by inertial forces

l l0 >> l >> ηϵ

ν

lEI l0 LlDIη

Inertial SubrangeDissipation

rangeEnergy-containing

range

Turbulent Energy Spectrum

• Energy spectrum provides the distribution of energy among the different length-scales

• Plotted as a function of wavenumber

• Shape deduced from theory ➡ confirmed by experiments

k =2π

l

Numerical Simulation of Turbulence

• Direct numerical simulation (DNS) ➡ Solve the Navier-Stokes equations on a computational grid

• DNS needs to resolve all time and length scales of turbulence ➡ Grid-width should be comparable to Kolmogorov length scale

➡ Computational domain proportional to integral length scale

• Number of grid points (in each direction) depends on Reynolds number

N / l0⌘

=) N / Re3/4

DNS Cost

• Timestep also proportional to Reynolds number ➡ Total time of integration comparable to integral time scale

➡ Time-step based on Courant-number restriction

!

!

!

!

• Total cost = number of grid points X number of time steps

u0�t

�x

⇡ 1 ⌧0 ⇠ l0/u0

Nt =⌧0

�t

⇠ l0/�x ⇠ Re3/4

Cost / Re

3

DNS Uses

• DNS is often used to test turbulence models (to be discussed next)

• However, DNS is often performed at low Re numbers ➡ Computational limitations

Isotropic Turbulence

Example DNS using 4096^3 grid points

• Performed at the Earth Simulation Center in Japan

• Estimated computational time ~ 240 hrs/eddy turnover time

• Computed on 2000 processors

Turbulence Modeling

• Direct simulation of governing equations is not practical ➡ Reduced descriptions sought

- Leads to turbulence modeling

• Two main approaches ➡ Reynolds averaged Navier-Stokes (RANS) approach

➡ Large eddy simulation (LES) approach

• Other approaches ➡ Partially-averaged NS (PANS)

➡ Hybrid RANS/LES

RANS Approach

• Main motivation ➡ For engineering applications, only statistics of the flow field

required

- E.g. average velocity and its variance

• RANS formulates equations for moments of the flow variables

• RANS formulation based on one-point one-time probability density function (PDF) of flow variables ➡ Key properties such as dissipation are two point quantities and

require modeling

Ensemble Averaging

• Consider a turbulent flow prescribed using a set of boundary and initial conditions ➡ These conditions are insufficient to describe the flow completely

➡ Each realization thus produces variations due to the chaotic nature of turbulence

• Ensemble averages are defined as

Ui = ⟨Ui⟩ + ui Yi = ⟨Yi⟩ + yi T = ⟨T ⟩ + T′

i

hUii =1

N

NX

j=1

U ji

RANS Profiles

eventually become wake vortices. Such transition to wakevortices is more clearly noted in Case IV (downstream tri-angle).

The most notable vortical structure in the JICF is theCVP. Since the CVP is considered as the source of jet mix-ing in the region beyond the near-field, it has been widelyexamined [? ] [REF]. The source of the CVP for case I canbe found on the side of the jet, where the hanging vorticescreate the initial entrainment that finally produces thekidney-shaped deformation of the jet downstream througha series of complex vortex folding processes (see Fig. 3 in[? ]). In contrast to that study, the Reynolds number usedhere is higher and a clear definition of the CVP beyond theinitial formation process is not possible. Fig. ?? shows themean passive scalar mass fraction for two di�erent cases.From the contour slices parallel to the jet axis, the ini-tial CVP formation can be determined. For instance, inthe circular jet case, the jet is deformed with entrainmenttowards the leeside of the jet.

The circular jet is the most studied JICF configuration.In the next sections, the di�erences between the circularjet and other exit shapes are discussed.

3.3.2. Case II : Square jetThe square jet is fundamentally di�erent from the cir-

cular jet due to the alignment of the jet sides with thecrossflow. Due to this alignment, the hanging vortices ofthe square jet are more stable compared to those in thecircular jet. In addition to the DSR and hanging vortices,a small column vortex that entrains the crossflow into theleeside of the jet is observed (Fig. 8). This vortical struc-ture is highly intermittent and is located below the hangingvortex. In addition, the column vortex is not symmetri-cal. Although two column vector can be present on thetwo sides of the jet, at any time only one jet is found.The wake vortices emanating from the jet and ending inthe boundary layer on the leeside of the jet is strongerand more frequently observed compared to the circularjet. This structure is very similar to the Karman vortexstreet observed in flow around a cylinder.

- this part was not clear, look at the previous para. Ihave put in whatever makes sense - you need to add state-ment concerning the other parts. The jet velocity profileis responsible for the orientation of the vorticity that rollsoutward. For square jet, the jet edges are oriented prop-erly to the streamwise and spanwise vorticity, �x and �z

respectively. Since the side of the jet is aligned with thecrossflow, the hanging vortices of square jet is more stablethan the ones of circular jet. Although the vortical struc-tures in the downstream side of the jet are still chaotic,three distinct sources of the vorticity were observed. First,the downstream side of the jet creates DSR (positive �z).Secondly, remaining part of the jet is rolled inside (positive�y for positive z). Just below the hanging vortex on Fig. 8,there is a small column of that induces the crossflow intothe lee-side of the jet column. The presence of this small

Figure 7: Mean passive scalar field comparison of circular(left) and triangular (right) jet. White curve representsthe mean trajectory. Slices shown here are z/D=0, y/D=0,y/D=0.6, and x/D=5

column is very brief and random. Unlike hanging vor-tex, only one side of the jet was observed at a time. Ontwo dimensional flow field slice, this semi-periodical vortexmovement resembles Karman vortex street of flow arounda cylinder. Thirdly, separation of the crossflow boundarylayer [? ] which causes wake vortices takes place. Theorientation the wake vortices is consistent with Yuan etal.[? ].

3.3.3. Case III : Triangular (upstream) jetIn the upstream jet configuration, the apex of the tri-

angle is located upstream of the base. In terms of jettrajectory evolution, the upstream triangular jet providedthe highest mean trajectory and slightly faster jet break-down. Although the improvement was minimal, the near-field vortex evolution is very di�erent from the previous

7

therefore more variance along the trajectory is measuredin the near-field.

Figure 14: Instantaneous passive scalar contour on the sym-metric plane for parabolic (left) and top-hat(right) jet

4.3. Evolution of vorticityFrom the previous observation, parabolic jet has higher

trajectory and more e⇥cient mixing than top-hat jet. Sincevortices are important factor for the large scale flow cir-culation, its evolution for each case is studied carefully inorder to understand the di�erence.

First, parabolic velocity profile provides suitable con-dition for development of ”vortex shield” at the leadingedge of the jet flow in the near-field. The velocity gradientis not su⇥cient to create vortex that is strong enough todeflect the jet immediately, the jet is more upright thantop-hat jet. As the leading side of the jet column is ex-posed to the crossflow, the shear layer becomes thinner andthe more unstable, developing the ”vortex shield”. Fromthis shear layer thinning process, the magnitude of vor-ticity continuously increases, and the resultant instability

Figure 15: Instantaneous iso-surface of vorticity contouredwith passive scalar for parabolic (left) and top-hat(right)jet

cause the tip of the vortex shield to be dispersed into sev-eral vortices with extreme randomness. The surface areaof jet/crossflow interaction is increased dramatically, pro-viding a favorable condition for mixing in molecular level.

Top-hat jet’s velocity gradient is concentrated aroundthe jet edge, therefore strong vortex ring is developedaround the jet exit. The magnitude of vorticity is strongenough to entrain the crossflow immediately after the jetis exposed to the crossflow. As the result, the large vor-tical structure, especially leading-edge and lee-side vor-tex, is created at earlier stage. Although it entrains moreamount of the crossflow initially, the mixing solely relies onthe circulation in the core of the vortex ring. Compare tothe turbulent mixing, it is very ine⇥cient and slow. Thevortex-ring-like structure is formed periodically becauseonce it deflects the jet flow, the velocity gradient aroundthe jet is relieved allowing the jet to be upright for a while,

11

DNS RANS

Favre-Averaging

• For variable density flows, Reynolds-averaging leads to terms with fluctuating density

• Instead, Favre averages are defined

⟨ρUT ⟩ = ⟨ρ⟩⟨U⟩⟨T ⟩ + ρ⟨uT ′⟩ + ⟨ρ′u⟩⟨T ⟩ + ⟨ρ′T ⟩⟨U⟩ + ⟨ρ′uT ′⟩

U = !U + u′′ ⟨ρu

′′⟩ = 0

⟨ρU⟩ = ⟨ρ!U⟩ + ⟨ρu′′⟩ = ρ!U

⟨ρUT ⟩ = ρ!U⟨T ⟩ + ρ "u′′T ′′

Reynolds Stresses

• Turbulence modeling essentially refers to closure models for the Reynolds stress tensor

• Arise from the nonlinear term in the momentum equation

!

!

!

!

• Reynolds stresses cannot be obtained directly from the mean velocity field

• Signifies transfer of momentum by fluctuating field

∂ρUi

∂t+

∂ρUjUi

∂xj

= −

∂p

∂xi

+∂τij

∂xj

,

hUiUji = hUiihUji+ huiujiReynolds stress

Properties of Reynolds Stresses

• Second-order tensor ➡ Symmetric

➡ Diagonal terms are normal stresses, off-diagonal are shear stresses

➡ Turbulent kinetic energy is half the trace of tensor

!

• Two approaches to closing Reynolds stresses ➡ Turbulent viscosity based models

➡ Reynolds-stress transport equations

k =1

2huiuii

Turbulent Viscosity based Models

• requires closure

• Most common closure approach is based on the turbulent-viscosity hypothesis ➡ Introduced by Boussinesq (1877)

➡ Relates the deviatoric part of the Reynolds stress to the mean rate of strain

➡ Need models for and

- Several models have been proposed

⟨uiuj⟩

−ρ⟨uiuj⟩ = µt

!

∂⟨Ui⟩

∂xj

+∂⟨Uj⟩

∂xi

"

−2

3ρδijk

k =1

2⟨ujuj⟩

µt k

Example: Turbulent Round Jet

• Round jet issuing into quiescent surroundings ➡ Exhibits self-similar behavior

CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

fluidsupply

nozzled

U θ

r

WU

V

xquiescentsurroundings

J

Figure 5.1: Sketch of a round jet experiment, showing the polar-cylindrical coordinate system employed.

1

CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

0 10 200.0

0.1

0.2

〈U〉/UJ

x/d = 30

x/d = 60

x/d = 100

r/dFigure 5.2: Radial profiles of mean axial velocity in a turbulent roundjet, Re = 95, 500. The dashed lines indicate the half-width, r 1

2(x),

of the profiles. (Adapted from the data of Hussein et al. (1994).)

2

Round Jet: Self-similar Profile

• Normalized based on half-width of jetCHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

0 10 200.0

0.1

0.2

〈U〉/UJ

x/d = 30

x/d = 60

x/d = 100

r/dFigure 5.2: Radial profiles of mean axial velocity in a turbulent roundjet, Re = 95, 500. The dashed lines indicate the half-width, r 1

2(x),

of the profiles. (Adapted from the data of Hussein et al. (1994).)

2

hU(x, r1/2, 0)i =1

2hU(x, 0, 0)i

CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

0 10 200.0

0.1

0.2

〈U〉/UJ

x/d = 30

x/d = 60

x/d = 100

r/dFigure 5.2: Radial profiles of mean axial velocity in a turbulent roundjet, Re = 95, 500. The dashed lines indicate the half-width, r 1

2(x),

of the profiles. (Adapted from the data of Hussein et al. (1994).)

2

CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

0.5

1.0

0.00.0 1.0 2.0

r/r1/2

UU0

Figure 5.3: Mean axial velocity against radial distance in a turbulentround jet, Re ≈ 105; measurements of Wygnanski and Fiedler(1969). Symbols: ◦, x/d = 40; △, 50; !, 60; ♦, 75; •, 97.5.

3

CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

0 40 80 1200

5

10

15

20

25

x/d

UJU0(x)

Figure 5.4: Centerline mean velocity variation with axial distance in aturbulent round jet, Re = 95, 500: symbols, experimental data ofHussein et al. (1994), line—Eq. (5.6) with x0/d = 4, B = 5.8.

4

Reynolds Stresses in Round Jet

• Stress tensorCHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

0.0 1.0 2.0 3.00.00

0.02

0.04

0.06

0.08

0.10

〈uiuj〉U02

r/r1/2

〈u2〉

〈v2〉

〈w2〉

〈uv〉

0.0 0.1 0.2 0.3η

Figure 5.7: Profiles of Reynolds stresses in the self-similar round jet.Curve fit to the LDA data of Hussein et al. (1994).

7

CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

0.0 1.0 2.0 3.00.0

1.0

2.0

3.0

4.0

5.0

〈U〉

r/r1/2

0.0 0.1 0.2 0.3η

Figure 5.8: Profile of the local turbulence intensity—⟨u2⟩12/⟨U⟩—in

the self-similar round jet. From the curve fit to the experimentaldata of Hussein et al. (1994).

8

2

4hu2i huvi 0huvi hv2i 00 0 hw2i

3

5

Turbulent Viscosity

• Turbulent viscosity is also self-similar

!

• Turbulent viscosity in terms of length scaleCHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

0.0 1.0 2.00.00

0.01

0.02

0.03

r/r1/2

νT^

0.0 0.1 0.2η

Figure 5.10: Normalized turbulent diffusivity ν̂T (Eq. (5.34)) in theself-similar round jet. From the curve fit to the experimental dataof Hussein et al. (1994).

10

⌫T (x, r) = U0(x)r1/2(x)b⌫T (⌘)

⌫T = u0l

CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

0.0 1.0 2.00.00

0.05

0.10

0.15

r/r1/2

r1/2l

0.0 0.1 0.2η

Figure 5.11: Profile of the lengthscale defined by (Eq. (5.35)) in theself-similar round jet. From the curve fit to the experimental dataof Hussein et al. (1994).

11

Transport of Kinetic Energy

• Transport of kinetic energy is important to understand for modeling purposes

• Derived from governing equations (no approximations here)

!

!

!

!

!

• Production and dissipation terms are important

@k

@t

+@hUiik@xi

+@Ti

@xi= P � ✏

Ti =1

2huiujuji+ huip

0i/⇢� 2⌫hujsiji

sij =1

2

✓@ui

@xj+

@uj

@xi

◆

P = �huiuji@hUii@xj

✏ = 2⌫hsijsiji

Budget of Kinetic Energy Equation

• Production and dissipation

!

!

• Turbulent transport

!

!

• Mean-flow convection

CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

Cambridge University Press, 2000

c⃝Stephen B. Pope 2000

Production

Mean-flowconvection

Turbulenttransport

Dissipation

0.0 1.0 2.0r/r1/2

0.00

0.01

–0.01

–0.02

0.02

Gain

Loss

Figure 5.16: Turbulent kinetic energy budget in the self-similar roundjet. Quantities are normalized by U0 and r1

2. (From Panchapake-

san and Lumley (1993a).)

16

P = �huiuji@hUii@xj

✏ = 2⌫hsijsiji

@Ti

@xi

Ti =1

2huiujuji+ huip

0i/⇢� 2⌫hujsiji

@k

@t

+@hUiik@xi

Model for Kinetic Energy Production• Turbulent viscosity hypothesis

!

!

➡ Relates anisotropy tensor to strain rate

• Production related to anisotropy tensor

!

!

!

!

• According to the model, production is always positive

aij = huiuji �2

3k�ij = �⌫T

✓@hUii@xj

+@hUji@xi

◆

P = �aijSij Sij =1

2

✓@hUii@xj

+@hUji@xi

◆

aij = �2⌫TSij

P = 2⌫TSijSij

Transport Equation for Kinetic Energy

• Turbulent transport modeled using gradient diffusion hypothesis

!

!

!

• Modeled kinetic energy equation

!

!

➡ Dissipation rate still needs to be modeled

- Focus now shifts to modeling this term

Ti =1

2huiujuji+ huip

0i/⇢� 2⌫hujsiji

Ti = �⌫T

�k

@k

@xi, �k = 1.0

@k

@t

+@hUiik@xi

=@

@xi

✓⌫T

�k

@k

@xi

◆+ P � ✏

One-equation Model

• related to turbulent viscosity using mixing-length

➡ Coefficient is roughly 0.55

➡ is the mixing length and needs to be specified by the user

➡ needs to be obtained separately

➡ nows needs to be modeled

- Represents energy dissipation rate

k

lm

k

∂ρk

∂t+

∂⟨Uj⟩k

∂xj

=∂

∂xj

!

[µ + µt]∂k

∂xj

"

+ µt

!

∂⟨Ui⟩

∂xj

+∂⟨Uj⟩

∂xi

"

∂⟨Ui⟩

∂xj

− ϵ

ϵ

✏ =CDk3/2

lm

⌫T = ck1/2lm

Turbulence Model

• Dissipation rate can be solved using a transport equation ➡ Equation based on very strong closure assumptions

➡ Several modeling constants

k − ϵ

σϵ = 1.3 Cϵ1 = 1.44 Cϵ2 = 1.92

Pk = µt

!

∂⟨Ui⟩

∂xj+

∂⟨Uj⟩

∂xi

"

∂⟨Ui⟩

∂xj

µt = ρCµ

k2

ϵ

∂ρϵ

∂t+

∂ρ ⟨U⟩j ϵ

∂xj=

∂

∂xj

!

µt

σϵ

∂ϵ

∂xj

"

+ Cϵ1Pkϵ

k− ρCϵ2

ϵ2

k

Turbulence Model• In most RANS viscosity-based models, kinetic energy is set as

one variable ➡ A second variable is needed to obtain length scale

➡ Several choices possible

• Treat as the additional variable

!

!

➡ Expression for turbulent viscosity same as before

➡ Numerical issue when treating cases with non-turbulent regions (k=0)

k � !

@!

@t

+@hUii!@xi

=@

@xi

✓⌫T

�!

@!

@xi

◆+ C!1

P!

k

� C!2!2

! = ✏/k

Reynolds-Stress Transport Models

• Alternate approach to turbulent-viscosity methods

• Directly solves transport equations for the Reynolds stress tensor

!

!

• Production term (P) appears closed

• Turbulent transport (T), pressure-rate-of-strain tensor (R) and dissipation tensor ( ) need to be modeled

• Dissipation rate still needed ➡ Modeled using equation similar to the k-e model

➡ Except that the production term is directly obtained from the Reynolds stresses

@huiuji@t

+@hUkihuiuji

@xk+

@

@xkTkij = Pij +Rij � ✏ij

✏ij

Reynolds Stress Model

• Directly solves transport equations for the Reynolds stresses ➡ No need to use turbulent-viscosity hypothesis

➡ In three-dimensional flows, six additional equations have to be solved

➡ In addition, a transport equation for dissipation has to be solved

➡ Found to be more accurate than the two-equation closures for a variety of complex flows

- Swirling flows, flows with stagnation points etc.

➡However, many closures required leading to higher degree of model uncertainty

Issues in RANS Modeling

• RANS performance highly sensitive to turbulence models ➡ Models are based on simple flows

➡ Coefficients are tuned based on canonical flows and are not universal

➡ Modeling burden is too high with information only on mean quantities available

- In spite of decades of developments, models are still not sufficiently accurate for complex flows

• RANS unable to utilize the growing power of computers ➡ RANS accuracy is not related to the availability of computing

power