By: Ritthik Bhattacharya. 05NA3001. Under:

1. A new formulation for turbulent eddy viscosity based on anisotropy-I equation model for inhomogeneous flows. By: Ritthik Bhattacharya. 05NA3001. Under: Prof.Hari Warrior.

2. The objective: • Traditionally the effect of reynold stresses on the mean flow has been captured by the use of eddy viscosity. This eddy viscosity is calculated using the stability function approach. The work done here is to find out eddy viscosity using a different approach which uses the second invariant of anisotropy. The effect of inhomogeneity on the eddy viscosity near the boundary of the flow is also shown.

3. Points to touch upon: • Importance of turbulence. • Turbulence modeling. • Reynolds Stress model. • Formulation based on invariant of anisotropy. • The idea. • Importance of inhomogeneity. • Final model. • Results.

4. Importance of turbulence: Turbulence is all around us: • Air flowing in and out of lungs. • The natural convection in the room in which we sit. • Winds and gusts. • Drag on aeroplanes ,ships , bridges and buildings. • Atmospheric and oceanic flows. • Even the liquid core of earth.

5. Turbulence Modeling: • Fluid motion is governed by two main equations: 1.Equation of continuity: div (u)=0 for incompressible flow. u: velocity of fluid. This is basically a statement for conservation of mass.

6. 2.Navier Stokes Equation: ρ*(Du/Dt) =–grad p + ρ*ν*div2(u) which essentially means that the rate of change of momentum of a fluid element is equal to pressure gradient force plus the frictional force in absence of body forces. Here the frictional force is modelled assuming newtonian fluid.

7. NS equation in terms of vorticity:

8. Now the navier stokes equation for velocity leads to a velocity distribution with time as: u=U+u’ u: total velocity. U: Mean velocity. u’: randomly fluctuation velocity .

9. Putting U+u’ in place of u makes the navier stokes equation the reynolds stress equation: ρ*(DU/Dt) =–grad P + ρ*ν*div2(U) +ρ div( reynold stress). Reynold stress being of the form: <-u’I u’j > That is the TIME AVERAGE of product of fluctuating velocity components.

10. Closure problem: • When the equation is deterministic the velocity profile has a mean and fluctuating part. • Now when Reynold equation gives a formulation for the well behaved mean velocity then it contains statistical terms which don’t have exact governing equation!!

11. This is called the closure problem. • The whole point of turbulence modeling is to model the behaviour of these statistical terms using empirical equations. • Essentially it is a sophisticated exercise in interpolating between data sets.

12. Types of Turbulence models: • Eddy viscocity models. In these models the reynolds stresses are assumed to have the same effect as the mean shear stresses just with an augmentation by a factor vt /v. v is the coefficient of molecular kinematic viscosity while vt represents a much higher “eddy viscosity”.

13. -<u’I u’j > =2vt Sij –(2k/3)*δij Sij :Mean shear stress. δij : Dirac Delta function,such that the sum of tarces of reynolds stress be equal to -2k.(k:being the kinetic energy). δij=1 for(i =j) =0 otherwise

14. One equation models. In these models apart from the Reynolds equations another transport equation for turbulent kinetic energy is used.

15. Two equation models: In addition to the reynolds equation two additional transport equations are used One for turbulent kinetic energy and the other for energy dissipation (ε) is used.

18. where Prt is the turbulent Prandlt number for energy and gi is the component of the gravitational vector in the ith direction. For the standard and realizable - models, the default value of Prt is 0.85. β is the coefficient of thermal exapnsion.

19. Reynolds Stress Models. • In reynolds stress models the two transport equation used in addition to the reynolds averaged equation are the one for reynolds stresses and another for rate of dissipation of kinetic energy.(ε)

20. The equation for reynolds stresses: d<u’I u’j>/dt= Pij+Rij-εij+dij Here: Pij is the production of reynold stress modelled by

21. This is equal to: -(4/3)*k*Sij • Rij is the pressure rate of strain term which distributes reynolds stress values. It is divided into two terms: 1.Slow pressure rate of strain. (Rijs) 2.Rapid pressure rate of strain.(Rijr)

22. Rate of slow pressure strain is important in the process of return to isotropy, where the turbulence decays thus reverting from its anisotropic state to isotropy. When a system is excited into turbulence through the effect of mean shear or buoyancy, the turbulence develops into an anisotropic mode. When such an anisotropic turbulence is allowed to decay the system returns to isotropy.

23. Rotta proposed the famous linear model for the return to isotropy. Rijs=-2CRεaij Where CRis rotta’s constant, while aij is the invariant of anisotropy defined as aij=<u’I u’j >/2k - δij/3

24. The rapid pressure strain rate is modeled as: Rijr= -0.6*( Pij- Pii*δij/3 ) For inhomogeneous flows.

25. εij is the dissipation term for reynolds stress. For high reynolds number flow it is found to be: εij =(2/3)εδij Where ε is the energy dissipation rate.

26. dij is the transport term which is non zero for inhomogeneous flows only. It consists of the divergence of a term. dij= ∂/ ∂(xk) [{.22 k2/ε} ∂/ ∂(xk) (<u’iu’j>)]

27. The transport equation for ε is nearly the same as that used in the k- ε model with an anisotropic coefficient used instead of a constant in the transport term.

28. The invariant of Anisotropy. • We have already defined, aij=<u’iu’j>/2k - δij /3 and the eddy viscosity hypothesis as: <-u’iu’j>=2vtSij – (2k/3) δij Therefore using the two above: aij = -vt*Sij/k

29. We define the invariant of anisotropy Π=aij aji Using the formulation of aij , Π =(vt)2 *(Sij)2/k2 so that,

30. s=√(Sij)2 • This is the important step. This means that the eddy viscosity can be calculated at any point given that we find the value of the invariant of anisotropy at that point and we find the mean velocity components there.This method needs less assumptions than the stability function approach estimating, vt=Ck2/ε

31. C being the stability function containing all the nonlinearity of the problem.

33. For isotropic 2 D turbulence Π=1/6 i.e vt=(1/√6)*k/S For isotropic 3D turbulence Π=0 i.e vt=0 For 1 D turbulence Π=2/3 i.e vt= √(2/3)*k/S

34. The idea was to: • Use the formulations for the transport of reynolds stress in RSM and that of k from k-ε model and use them to find a transport equation for Π which can give the Π value for all times and hence the value of vt for all times.

35. This was performed by Craft et al. for homogenous flows.The addition i made was for inhomogeneous flows.

36. The importance of inhomogeneous flows: • The viscous sub layer of flow for a flow near a boundary is inhomogeneous that is the spatial gradients of the turbulent quantities can NOT be neglected in the viscous sublayer. • The boundary of a jet or wake where entrainment of ambient flow occurs into the turbulent wake field is inhomogeneous.

37. We start from: 1.Dk/dt=d + P –ε from k- ε model, d: transport of turbulent kinetic energy. P: production of kinetic energy. P=-aij *k*Sij ε:dissipation of turbulent KE

38. 2.d<u’I u’j>/dt= Pij+Rij-εij+dij Using these two we get governing equation for aij. Multiplying the expression for daij./dt by aij. we get expression for dΠ/dt.

39. Therefore the equation for Π becomes: • In the homogenous case the expression for dΠ/dt is found out to be:

40. In the inhomogeneous case two extra terms need to be modeled d and dij. • Since the effect of inhomogeneity is more on the reynold stress rather than the turbulent kinetic energy. So d is taken to be 0 and dij is modeled. The term added to dΠ/dt is 2aij*dij/k

41. dij =(.22k2/ε)[∂2/ ∂xk ∂xk (<u’iu’j>)] Since we are concerned with shear flow in i direction so spatial derivatives wrt xi is negligible. So the only spatial derivatives taken into consideration are those wrt xj.

42. Therefore dij=(.22k2/ε)[∂2(aij)/ ∂xj ∂xj] *2k There for the term to be added, (.44k2/ ε)[aij ∂2(aij)/ ∂xj ∂xj] = (.44k2/ ε) [[∂{aij ∂(aij)/ ∂xj}/ ∂xj]-(∂(aij)/ ∂xj)2] = (.44k2/ ε)[.5*∂2(Π)/ ∂xj2 –(∂(√ Π)/ ∂xj)2] The derivatives are calculated using finite difference schemes

43. Results: • Though the results for the eddy viscosity was not decisively better than the previous stability function approach it was arrived using much lesser assumptions. Also the inhomogeneous term changed the value of eddy viscosity near the boundary of fluid flow in the POM.

45. The nature of change of eddy viscosity with depth and time varies nearly the same way as in the stability function model

48. Thank You!!