2. Turbulence Modeling In FLOTRAN Chapter 5

3. Training Manual May 15, 2001 Inventory #001478 5-2 Questions about Turbulence What is turbulence and what are its characteristics? Why is it important? How are the basic equations affected? –Time Averaging –Navier-Stokes Eddy Viscosity Models –Simple Approaches (FLOTRAN) –One Equation Models –Two Equation Models (FLOTRAN) Wall Treatments –Prandtl’s approach –Log -Law of the Wall –How to Treat Wall Roughness

4. Training Manual May 15, 2001 Inventory #001478 5-3 What Is Turbulence? Fluid State characterized by agitation, disorder, disturbance. It is caused by shear forces which act to disrupt an otherwise orderly flow The result is that the velocity of a “steady flow” is no longer exactly constant Velocity vs. Time looks as follows:

5. Training Manual May 15, 2001 Inventory #001478 5-4 General Flow Characteristics Bulk Flow Velocity –Length Scale associated with velocity Large Scale Eddies –Visible Flow features Fine Flow Structure –Smallest length scale Dissipation Eddies –Turbulent energy is returned to the “main flow” Separation Time Variant Flow

6. Training Manual May 15, 2001 Inventory #001478 5-5 Importance of Turbulence Heat Transfer Enhancement Pressure Drop Pressure distribution –Lift –Drag Affects Separation

7. Training Manual May 15, 2001 Inventory #001478 5-6 Time Averaging The disorder caused by the shearing forces gives rise to a time-varying velocity. –Bulk steady state problems, e.g. River Flows, are locally transient in nature Separate the Velocity into a Mean (steady state average) Component and a Fluctuating Component. Put this new expression for the velocity into the governing equations and integrate over a time interval just long enough for the fluctuations to “average out”. The result are the Reynolds Averaged Navier Stokes Equations Now for the procedure….

8. Training Manual May 15, 2001 Inventory #001478 5-7 Time Averaging the Velocity Velocity = Mean Value + Fluctuating component t

9. Training Manual May 15, 2001 Inventory #001478 5-8 The Time Averaging Process Expression for Velocity Time Average of Fluctuation Component Integrates to zero by definition. Time Average of the complete velocity term integrates to the average velocity by definition.

10. Training Manual May 15, 2001 Inventory #001478 5-9 Each of the terms in the Navier Stokes Equations is time integrated Substitute the expression for actual velocity into the equations The time integration will involve integration's of the spatial derivatives…. Time Average of the Navier Stokes Equations

11. Training Manual May 15, 2001 Inventory #001478 5-10 Integrate the Spatial Derivative The derivative is over space; it is removed from the integral, so the term is zero

12. Training Manual May 15, 2001 Inventory #001478 5-11 The Diffusion Terms Expanding, the diffusion terms: By the relationship on the previous page, integration of the last term will yield zero and so there are no additional terms resulting from the time average of the diffusion portion.

13. Training Manual May 15, 2001 Inventory #001478 5-12 Now do the advection term –Substitute expression for velocity : –First term on the RHS comes from the average velocity Advection Terms

14. Training Manual May 15, 2001 Inventory #001478 5-13 Integrating Product terms... No contribution !!

15. Training Manual May 15, 2001 Inventory #001478 5-14 Only the final term provides a contribution: Advection Contribution

16. Training Manual May 15, 2001 Inventory #001478 5-15 Reynolds Averaged Navier Stokes Equations I and J range from 1 to 3 (E.G. X,Y,Z) There are three momentum equations There are three Reynolds Stress terms for each momentum equation –Nine additional terms reduce to six through symmetry Turbulence Closure Problem: Evaluate These Six Terms !

17. Training Manual May 15, 2001 Inventory #001478 5-16 Evaluation of Reynolds Stresses Reynolds Stress Models –Develop transport equations for the Reynolds Stress terms Algebraic Stress Models –Convert the transport equations for the Reynolds Stresses into algebraic equations Eddy Viscosity Models –Assume that the Reynolds Stresses are proportional to the mean velocity gradients. –The resulting proportionality factor must then be evaluated

18. Training Manual May 15, 2001 Inventory #001478 5-17 Eddy Viscosity Model General linear stress-strain relation for turbulent stresses: This is completely analogous to the laminar stress-strain relation: –Hence the term Reynolds Stress Additional terms introduced: –  Turbulent viscosity – k Turbulent kinetic energy

19. Training Manual May 15, 2001 Inventory #001478 5-18 The Result In the Eddy Viscosity approach, the time averaged Navier- Stokes Equations revert to their original form: –“Actual” velocity replaced by the Mean velocity –Laminar viscosity supplemented by turbulent viscosity

20. Training Manual May 15, 2001 Inventory #001478 5-19 The New Quantities Turbulent Viscosity: Empirical Constant and length scale “ l” Turbulent kinetic energy:

21. Training Manual May 15, 2001 Inventory #001478 5-20 Zero Equation Model A simple expression for the turbulent viscosity is used which replaces the kinetic energy with derivative of the mean velocity Advantage: simplicity Disadvantages –Evaluation of the length scale –Empirical incorporation of curvature, buoyancy, shear, rotational effects –No modeling of transport of turbulence effects

22. Training Manual May 15, 2001 Inventory #001478 5-21 Two Equation Models The length scale is eliminated in favor of the kinetic energy and its dissipation rate: Turbulent kinetic energy as previously defined The Dissipation rate of kinetic energy is defined as

23. Training Manual May 15, 2001 Inventory #001478 5-22 Equation for Kinetic Energy Dissipation Rate equation Expression resulting for turbulent viscosity Spalding and Launder Model

24. Training Manual May 15, 2001 Inventory #001478 5-23 Turbulent kinetic energy Turbulent dissipation with More General K-E Transport Expressions

25. Training Manual May 15, 2001 Inventory #001478 5-24 The coefficient C  directly enters the generation of turbulence Generation of Turbulent Kinetic Energy The Production term.

26. Training Manual May 15, 2001 Inventory #001478 5-25 Model Variations Standard k-e-model Alternative k- e -models... empirical, coefficient adjusted to simple shear flows... semi-empirical, yield physically realistic values even for complex flows Desire less generation of turbulence in regions high velocity gradients for more accuracy... Note: C: is inversely proportional to the velocity gradients

27. Training Manual May 15, 2001 Inventory #001478 5-26 How to characterize turbulence? non-dimensional invariant of the deformation and rotation tensor with S IJ is the deformation tensor W IJ is the rotation tensor Bases of Alternative K-E Models

28. Training Manual May 15, 2001 Inventory #001478 5-27 Invariants The invariants can thus be interpreted as the mean deformation or rotation a turbulent eddy experiences during its life-time Turbulent time scale = mean eddy life-time

29. Training Manual May 15, 2001 Inventory #001478 5-28 Alternate Model Variations of Cμ SZL Model NKE Model

30. Training Manual May 15, 2001 Inventory #001478 5-29 Modified dissipation equation Standard model RNG-model: NKE-model:

31. Training Manual May 15, 2001 Inventory #001478 5-30 Alternative Models - When to Use? When regions of high turbulence generation are encountered. See if the new models provide significant reduction. –Swirl –Shear –Separation –Axial Acceleration

32. Training Manual May 15, 2001 Inventory #001478 5-31 Conditions Near the Wall When viscosity is present, there is no flow at the wall –No Slip Condition The velocity profile near the wall is important. –Pressure Drop –Separation –Shear Effects –Recirculation Examination of experimental data yields a wide variety of results in the boundary layer …..

33. Training Manual May 15, 2001 Inventory #001478 5-32 Velocity Profiles Near the Wall

34. Training Manual May 15, 2001 Inventory #001478 5-33 Prandtl’s Approach to the Wall Denoting with a “prime” the fluctuating components, begin with the expression for the Reynold’s Stress: Assume that the fluctuating components are proportional to each other: Prandtl then proposed that the distance traversed by a fluid particle before reaching the mean velocity of the fluid is proportional to the magnitude of the fluctuating component

35. Training Manual May 15, 2001 Inventory #001478 5-34 Prandtl …... Next assume that the mixing length depends on the distance from the wall: The resulting expressions for the shear stress become:

36. Training Manual May 15, 2001 Inventory #001478 5-35 Prandtl’s Approach to the Wall Take the square root: Rearrange: Define the shear velocity:

37. Training Manual May 15, 2001 Inventory #001478 5-36 Prandtl’s Log Law of the Wall Integrate and use the definition of the shear velocity: Introduce dimensionless velocity and distance:

38. Training Manual May 15, 2001 Inventory #001478 5-37 Prandtl’s Law of the Wall We put the equation in terms of the dimensionless variables Since There results

39. Training Manual May 15, 2001 Inventory #001478 5-38 Prandtl’s Law in FLOTRAN Prandtl’s student Nikuradse evaluated the constants based on observation of data: The constant can be introduced into the log term using The FLOTRAN expression for the Log Law is thus:

40. Training Manual May 15, 2001 Inventory #001478 5-39 Log Law of the Wall

41. Training Manual May 15, 2001 Inventory #001478 5-40 Variations With the preceding data appropriate for smooth walls, it is natural to ask how wall roughness affects the profiles The effect is to lower the curve on the dimensionless u + vs. y + plot The additional term results in a difference in the Y intercept of the curve…...

42. Training Manual May 15, 2001 Inventory #001478 5-41 Log Law for Rough Walls

43. Training Manual May 15, 2001 Inventory #001478 5-42 Wall Roughness Parameters An additional parameter has been introduced Here “k” is the actual peak to valley height of the wall imperfection. An expression for the Y intercept change due to Prandtl and Schlichting is often used:

44. Training Manual May 15, 2001 Inventory #001478 5-43 The roughness in FLOTRAN will be handled by modifying the constant E (found under Wall Parameters in the GUI): So that Where Wall Roughness in FLOTRAN Terms

45. Training Manual May 15, 2001 Inventory #001478 5-44 Roughness - An Example Consider the flow of water through a Cast Iron 12 inch pipe Take the roughness as : k/d = 8.0E-4 Density: 9.0x10 -5 (lbf-sec 2 )/(in 4 ) Viscosity: 1.5x10 -7 (lbf-s)(in 2 ) Velocity of 30 in/sec yields –Reynolds Number ~2.1x10 5 What value of E should we use in FLOTRAN?

46. Training Manual May 15, 2001 Inventory #001478 5-45 Roughness Example Strictly speaking, the process is an iterative one, since E is a function of k+ which in turn is a function of the unknown shear stress. It should be possible to achieve a good initial approximation however, using the smooth wall shear stress. The results can be checked with the Moody Chart, which provides data relating shear stress, relative roughness, and Reynolds Number.

47. Training Manual May 15, 2001 Inventory #001478 5-46 Roughness - Estimate E The results of an axisymmetric case with a 100 inch long pipe indicate that the smooth wall shear stress for developing flow tends towards the value: TAUW = 1.3x10 -4 Boldly double that value: use 2.6x10 -4 as a first guess. And thus the revised value of E

48. Training Manual May 15, 2001 Inventory #001478 5-47 The resulting shear stress from FLOTRAN is: Lets see what value of E that shear stress would imply… So that the new E would be: Roughness Example...

49. Training Manual May 15, 2001 Inventory #001478 5-48 More Roughness.. Continued iteration is possible, as are studies with mesh refinement.

50. Training Manual May 15, 2001 Inventory #001478 5-49 Wall Roughness - Prandtl-Schlichting Relation

51. Training Manual May 15, 2001 Inventory #001478 5-50 The Moody Chart

52. Training Manual May 15, 2001 Inventory #001478 5-51 Checking with Moody... For k/d of 8.0E-4: Therefore: Meanwhile So that with: Moody Chart Shear Stress is thus

53. Training Manual May 15, 2001 Inventory #001478 5-52 Final Check How close does the Moody Value of 2.58 check out? FLOTRAN result with E of 2.58 yield a shear stress of 1.7x10 -4 Results are within 20 percent.