Introduction to Mathematical Analysis of Turbulent Flows

Gautam Biswas Indian Institute of Technology Guwahati

Books:

A First Course in Turbulence H. Tennekes and J. Lumley

Introduction to

Fluid Mechanics and Fluid Machines S.K. Som, G. Biswas and S. Chakraborty

Mathematical Analysis of Turbulent Flows Highlighting Some Applications

http://www.iitk.ac.in/me/faculty/gautam.html

http://nptel.iitm.ac.in

Characteristic features of turbulence

• Irregularity Complex variations of velocity, temperature etc. with space and time (fluctuations)

The irregular motion is generated due to random fluctuations. It is postulated that the fluctuations inherently come from disturbances (such as, roughness of the solid surface) and they may be either damped out due to viscous damping or may grow by drawing energy from the stream. At a Reynolds number less than critical, the kinetic energy of flow is not enough to sustain the random fluctuations against the viscous damping and in such cases laminar flow continues to exist. At somewhat higher Reynolds number than the critical Reynolds number, the kinetic energy of flow supports the growth of fluctuations and transition to turbulence is induced.

• Strong mixing High transfer of momentum, heat, mass by fluctuating turbulent motion-

practically most important feature

• Turbulent motion always 3D For a parallel flow, it can be written that the axial velocity component is

y is the normal direction, г is any space variable

Even if the bulk motion is parallel, the fluctuation u’ being random varies in all directions. Now let us look at the continuity equation.

Since , the above equation depicts that y and z components of velocity exist for the parallel flow if the flow is turbulent. We can write

( ) ( ) ( )

( )( )

, ,

0 ,

0 ,

u y t u y u t

v v y t

w w y t

′= + Γ

′= + ′= +

0ux′∂≠

∂

0u u v wx x y z

′∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂

( ) ( ) ( ), ,u y t u y u t′= + Γ

• Turbulent motion carries vorticity – is composed of eddies interacting with each other

• Wide spectrum of eddy sizes and corresponding fluctuation frequencies

Turbulence

1. Disorganized, Chaotic, Seemingly Random

Behaviour

2. Non-Repeatability

3. Enhanced Diffusion (mixing) and Dissipation

4. Three Dimensionality and Rotationality

5. Intermittency in both space and time.

Source:

www. cora.nwra.com/~werne

Source: www.nwtpas.ca/science - waterfalls

WHY DO WE NEED TURBULENCE MODELS 1

Is direct simulation of turbulence possible ?

• Exact equations are known:

• They can be solved numerically in principle • Flow field must be discretized • Main problem: at high Re, turbulence consists of wide spectrum of eddy sizes.

21 , 0i i i ijj i j j i

U U U UPUt x x x x x

νρ

∂ ∂ ∂ ∂∂+ = − + =

∂ ∂ ∂ ∂ ∂ ∂

= grid size = velocity change per grid N = number of grid points per coordinate direction • for computers used these days for numerical solutions. This results in for 3-dimensional flows. • This values yields: . Only for Reynolds numbers of this kind direct numerical simulations are possible.

• For higher Reynolds numbers direct solution of the Navier- Stokes equation

is not possible.

x∆U∆

100≈maxN610≈totalN

20000≤maxRe

WHY DO WE NEED TURBULENCE MODELS 2 • Prospects of large-eddy simulation

• Resolution of only larger eddies

• Effect of eddies smaller than mesh size simulated by subgrid-scale model (or by numerical damping)

• Can be applied to high-Re flows especially when viscous sublayer not resolved but bridged by wall functions

• Will be method for practical calculations in 5-10 years. • Large computing times to get statistics (∼100000 time steps/ samples)

Why do we need turbulence models 3

• Statistical calculation methods • Details of turbulent fluctuations usually not of interest to engineers anyway.

• Hence statistical approach is taken and turbulence is averaged out.

• Statistical quantities:

, , 1/

o

o

t T

i i it

U U u P P p U T Udt+

′ ′= + = + = ∫

Turbulent flows can only be “mentally digested” if time averaged properties are given. These are denoted with a cross bar over the variable. = time averaged velocity = turbulent velocity fluctuation in the j-direction The following decomposition is valid: The averaged values is: For the fluctuations:

jU

ju′

)t,x(u)x(U)t,x(Û ijijij ′+=

∫∞→=T

ijTij dt)t,x(ÛT)x(U

0

1lim

∫ =′=′ ∞→T

ijTij dt)t,x(uT)x(u

001lim

The phenomena that need to be described are sketched below.

The computation of the instantaneous velocity fields is usually not attempted (DNS is required for that) because too much information results for the entire flow field, i.e., for the variations in space and time. Time averaged quantities are of interest.

The continuity equation for ρ = constant is: (1) The time average of this equation yields: (2) Or rewritten: (3)

Hence, we can also deduce: (4) At every point in time the turbulent fluctuations have fulfilled equation (4) exactly.

( ) 00 =′+∂∂

→=∂∂

iiii

i uUxx

U

( ) 0lim0

=′+∂∂

∫∞→

dtuUx ii

T

iT

( ) 01lim0∫ =∂

∂=′+

∂∂

∞→

T

i

iiiTi x

UdtuUTx

0=∂′∂

i

i

xu

Instantaneous values: Averaging yields per definition: or This yields: In connection with averaging over products:

bBB̂&aˆ ′+=′+= AA

BdtB̂T

&AdtˆT

T

T

T

T=∫=∫

∞→∞→ 00

1limA1lim

Bˆ&ˆ == BAA

0b0a =′=′ &

baAbBaBA)bB()aA(BA ′′+′+′+=′+′+=ˆˆ

0AA0Ba =′=′=′=′ bb&Ba

ˆ ˆAB A B a b A B a b′ ′ ′ ′= + = +

The momentum equation reads: If one adds the continuity equation multiplied with One obtains: Again introduced into the equation yields:

jijiji

ji

j gˆxx

P̂x

ÛÛ

tÛ

ρ+τ∂∂

−∂∂

−=

∂∂

+∂∂

ρ

jÛ

ˆ ˆ( ) 0j ii

U Ut xρ ρ

∂ ∂+ = ∂ ∂

jijiii

jij gˆxx

P̂x

)ÛÛ(t

)Û(ρ+τ

∂∂

−∂∂

−=∂ρ∂

+∂ρ∂

iiijjj uˆuˆ ′+=′+= UUandUU

( )totalmomentum transport term

i j ij i j ji j

PU U u u gx x

ρ τ ρ ρ∂ ∂′ ′+ + = − +∂ ∂

The individuals steps to derive the averaged momentum equations: Furthermore: Yielding the averaging laws of slide 15 we obtain: = convective momentum transport

= turbulence caused momentum transport

= molecular momentum transport

( )∫ =∂∂

∞→

Tj

TdtÛ

tT 001lim ρ

( )( )[ ]( )

∫ =′+′+∂∂

′′+′+′+∞→

T

uuUuUuUU

jjiiiT

....dtuUuUxT

jiijjiji0

1lim

ρ

ρ

( )[ ] ji

ij

jjiji

ig

xxPuuUU

xρ

τρρ +

∂

∂−

∂∂

−=′′+∂∂

ji UUρ

jiuu ′′ρ

ijτ

Introducing simplifications arising out of continuity equation we shall obtain

The above equation looks like laminar flow equation with the exception that the velocities are average velocities and there are additional stresses arising out of the fluctuating components!

2

2

u u u u pu v w ut x y z x

u u v u wx y z

ρ ρ µ

ρ

∂ ∂ ∂ ∂ ∂+ + + = − + ∇ ∂ ∂ ∂ ∂ ∂

′ ′ ′ ′ ′∂ ∂ ∂− + +

∂ ∂ ∂

σT is the Reynolds stress tensor and written in compact form as

' 'i ju uρ−

2

2

2

xx xy xz

T xy yy yz

xz yz zz

u u v u w

u v v v w

u w v w w

σ τ τσ τ σ τ ρ

τ τ σ

′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′= = − ′ ′ ′ ′ ′ ′ ′ ′

22xx

xy

up ux

u v u vy x

σ µ ρ

τ µ ρ

∂ ′= − + −∂

∂ ∂ ′ ′= + − ∂ ∂

The quantities are the turbulence caused momentum transport terms (5) This tensor is symmetric and has 6 unknowns. Analogous to (for ρ = const.): (6) We introduce: (7) = “eddy viscosity” simplest way to evaluate this as

′′ρ′′ρ′′ρ

′′ρ′′ρ′′ρ

′′ρ′′ρ′′ρ

=′′ρ

332313

322212

312111

uuuuuuuuuuuu

uuuuuuuu ji

jiuu ′′ρ

ijτ

∂∂

+∂∂

µ−=τj

i

i

jij x

Ux

U

tµ

ijj

i

i

jtji kx

Ux

Uuu δµρ

32

+

∂∂

+∂

∂−=′′

The term involving the Kronecker delta δij is perhaps a somewhat unfamiliar addition to the eddy-viscosity expression, it is necessary to make the expression applicable also to normal stresses (when i = j). The first part involving the velocity gradients would yield the normal stresses.

Whose sum is zero because of the continuity equation. However, all normal stresses are by definition positive quantities, and their sum is twice the kinetic energy k of the fluctuating motion:

Inclusion of the second part of the eddy viscosity expression assures that the sum of the normal stresses is equal to 2k. The normal stresses act like pressure forces (i.e. perpendicular to the faces of a control volume), and because, like the pressure itself, the energy k is a scalar quantity, the second part constitutes a pressure.

23

jii j t ij

j i

UUu u k

x xν δ ∂∂′ ′− = + − ∂ ∂

' 2 '2 '2 31 21 2 1

1 2 3

2 , 2 , 2 ,t t tUU Uu v u v u v

x x x∂∂ ∂

= − = − = −∂ ∂ ∂

( )' 2 '2 '21 2 312k u u u= + +

•The derivations showed: • That time averaging the continuity equation and the momentum equations yields the following equations: (8)

(9)

• Equation (8) and (9) are also referred to as Reynolds equations. One sees that the correlation of the velocity fluctuations yield a new term in this equation. This term is usually referred to as “Reynolds shear stresses”. It represents the momentum transport caused by the turbulence fluctuation. This tensor contains 6 independent components. Additional equations are needed to solve flow problems.

jiuu ′′

0=∂∂

i

i

xU

i

jij

iiji

ji

j

xuu

Uxx

vxP

xU

Ut

U∂

′′∂−

∂∂∂

+∂∂

−=∂

∂+

∂

∂ 21ρ

Subtracting equation (8) and (9) from the first two equations in slide 18 yields:

(10) If one multiplies equation (10) with and derives the similar equation for and multiplies this with , the following equation for the Reynolds stresses can be derived: (11)

21j j j i j i ji i j

i i i i j i i

u u U u u u u pU u v ut x x x x x x xρ′ ′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂′ ′+ + + − = − +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

iu′ iu′

ijij Tk

kji

Pk

jki

k

ikj

k

jik

ji

xuuu

xU

uuxUuu

xuu

Utuu

∂

′′′∂−

∂

∂′′−

∂∂′′−=

∂

′′∂+

∂

′′∂

21 2

ij ijij

jij i i j

i j k k l l

D

uup pu u v v u ux x x x x xρ

∈Π

′∂′′ ′ ∂∂ ∂ ∂′ ′ ′ ′− + − + ∂ ∂ ∂ ∂ ∂ ∂

ju′

For i = j one obtains the equation for the turbulent kinetic energy (k = 0.5 , introducing also the following equations: (12) (13) k-equation: (14)

1 2j i i j i

i j i

p pu u u px x xρ ρ=

′ ′∂ ∂ ∂′ ′ ′ ′− + = − ∂ ∂ ∂

20.5 )i iu u q′ ′ =

2 2ji i ii jk k k k

uu u uv vx x x x=

′∂ ′ ′ ′∂ ∂ ∂− = − ∂ ∂ ∂ ∂

2

production turb.transport dissipation

U 1U i k i ik i k ik k k i k k l l

ku u uk k ku u u p v vt x x x x x x x xρ

′ ′ ′∂ ∂ ∂ ∂∂ ∂ ∂ ∂′ ′ ′+ = − − − − +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

The modelling of the turbulent transport terms can be carried out by postulating an analogy to molecular momentum transport: (15) Kolmogorov introduced (1941): (16) Hence, we obtain: (17) (18)

1/22 and3

jii j t ij t

j i

UUu u v k v k lx x

δ ∂∂′ ′− = + − ∞ ∂ ∂

3/2 3/2

;k kll

∈∞ ∞∈

3/2 21/2

tk kv k∞ ∞∈ ∈

0902

.c;kcvt ≅∈= µµ

(18)

0902

.c;kcvt ≅∈= µµ

jt i ii t

i i k i j i j

UU Uk k kUt x x x x x x

νν ν ε

σ

∂ ∂ ∂∂ ∂ ∂ ∂+ = + + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂

2

production turb.transport dissipation

U 1U i k i ik i k ik k k i k k l l

ku u uk k ku u u p v vt x x x x x x x xρ

′ ′ ′∂ ∂ ∂ ∂∂ ∂ ∂ ∂′ ′ ′+ = − − − − +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

Compare between the original equation and model equation!

=∂∈∂

+∂∈∂

ii x

ut

ndissipatio ofdiffusion Viscous

ndissipatio ofn destructioTurbulent 2

ndissipatio ofdiffusion Turbulent 12

production mixed 2

productiongradient 2

production turbulent2

gradientity mean veloc 2

2

222

2

∈∇+

Φ

∂∂′∂

∂∂′∂

−

∂′∂

∂′∂

∂∂

−

∂

′∂∂

′∂′∂∂

−

∂′∂

∂′∂

∂∂

−

∂′∂′

∂∂∂

−

∂′∂

∂′∂

∂′∂

−

∂∂

∂

′∂∂

′∂−

∈

∈

∈

ν

ν

ρν

ν

ν

ν

ν

ν

mk

i

mk

i

m

k

mk

m

i

m

ik

k

j

k

j

i

k

i

j

ik

jk

i

m

k

m

i

k

i

k

i

k

j

i

j

xxu

xxu

D

xu

xp

x

xu

xuu

x

P

xu

xu

xu

xuu

xxu

xu

xu

xu

bypxu

xu

xu

Є - Equation

Equation for • 10 additional terms:

- Production of dissipation

- Destruction of dissipation

- Molecular diffusion of dissipation

- Turbulent transport of dissipation

• 7 terms are unclosed ⇒ need closure for this terms

• Modelling of ε–equation just by “dimensionless” analysis, because no

experimental based closure possible with

∂∂

∂∂

+∈

−∈

=∈ ∈

∈∈∈

j

t

jk xxk

Ck

PCDtD

σµρρ

2

21

ji ik t

j i j

uu uPx x x

µ ∂∂ ∂

≈ + ∂ ∂ ∂

ε

Standard model constants:

Cμ = 0.09, C1ε – 1.44, C2ε = 1.92,σk = 1.0, σε = 1.3

Rf = Richardson number, C3ε = 0.8. In absence of buoyancy G = 0, C3ε = 0

jt i ii t

i i k i j i j

P

ti

t i

G

Uv U Uk k kU vt x x x x x x

gx

εσ

ν φβσ

∂ ∂ ∂∂ ∂ ∂ ∂+ = + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂+ ∂

( )( )2

1 3 21t

i fi i i

vU C P G C R C

t x x x k kε ε εε

ε ε ε ε εσ ∂ ∂ ∂ ∂

+ = + + + − ∂ ∂ ∂ ∂

• Wilcox (1988) and Speziale et al. (1990) regard ω as the ratio ε of and k. Here we present the model due to Wilcox as

• The turbulent viscosity is related to k and ω by the expression (19)

• The transport equations for turbulent kinetic energy (k) and its dissipation

rate per unit turbulence kinetic energy (ω) are

(20)

(21)

The coefficients have the following empirically derived values

*

j

i

i

j

j

it

ik

t

iii kx

Ux

UxU

xk

xxkU

tk βωρµ

σµµρρ −

∂∂

∂

∂+

∂∂

+

∂∂

+∂∂

=∂∂

+∂∂

,tk

kεµ ρ ω

ω= =

0.2,0.2,100/9,40/3,9/5 * ===== ωσσββα k

2ωρβωαωσµµωρωρω

−+

∂∂

+∂∂

=∂∂

+∂∂

ki

t

iii Pkxxx

Ut

Figure 1: Vector plots of secondary flow (Deb, Biswas et al, IJHMT, 38, 2427-2444, 1995): ReH/2 = 67000

Figure 2: Contours of streamwise velocity at different cross-sections for ReH/2 = 67000

• Across a turbulent boundary layer, the flow has to undergo a transition from fully turbulent to completely laminar within the thin viscosity-dominated sublayer adjacent to the solid surface.

• This phenomenon is termed as Low Reynolds number turbulence and the transition from high to low Reynolds number regions is determined by the local turbulence Reynolds number, Rt = ρ k2/ (µε) where k is the turbulence kinetic energy and ε is its dissipation.

• The significant physical effect of the presence of a wall is:

- The viscous diffusion terms which are usually negligible compared to other terms in regions away from wall, become one of the largest terms to be balanced by the other terms.

Special Features of Near Wall Flows

Structure of a Turbulent Boundary Layer Three distinct zones are identified • The Viscous Sublayer (0 < y+ < 12) • The log Layer (30 < y+ < f (Re)) where the inertial terms can still be neglected due to vicinity of wall but the Viscous Stress is also negligible compare to the Reynolds Shear Stress •The Defect Layer where the flow behaves like the far wake behind a non-streamlined body.

Near Wall Treatment in Transport Equation based Models Wall Function Approach • Based on the equilibrium consideration (Production of k = Dissipation of k), k and ε or ω at the near wall node (P) are prescribed as following: • where Cµ is a closure coefficient. However, the Friction Velocity is not known a priori and it is an outcome of the iterative type solution algorithm κ is von Karman Constant

( )ρττ /wu =

( ) ( )PPPPP yCkyuCuk κωκ µτµτ 4/12/132/12 /and// ==∈=

We will discuss the near wall Velocity Profile Related issues in the next slides

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