Numerical Heat and Mass Transfer

Transcription

1 Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 19 Turbulent Flows Fausto Arpino

2 Introduction All the flows encountered in the engineering practice become unstable above certain Reynolds number. While at low Reynolds number the flows are laminar, at higher Reynolds number the flows are observed to become turbulent. In the turbulent regime a chaotic and random state of motion develops in which the velocity and pressure change continuously with time within substantial region of fluids. Most of flows in engineering applications are observed to be turbulent and so the study of turbulent regime is not just of theoretical interest. The Reynolds number gives a measure of the relative importance of inertia forces (associated with convective effects) and viscous forces. O ρu u x O µ u x Related to inertia forces Related to viscous forces = ρ U L µ U L = ρ U L µ = Re

3 Introduction At values of the Reynolds number above the critic value, the motion becomes chaotic and intrinsically unsteady, even with constant imposed boundary conditions. A typical point velocity measurement may be like the one in the figure: Typical point velocity measurement in a turbulent flow Velocity fluctuations in a turbulent flow 3

4 Introduction Even in flows where the mean velocities and pressures vary in only one or two space dimensions, turbulent fluctuations always present a three-dimensional character. Visualization of turbulent flows reveals rotational flow structures, the turbulent eddies, with a wide range of length scales. Fluid particles separated by a long distance can be brought close together by the eddying motions in turbulent flows; this aspect significantly enhance the heat, mass and momentum exchange process. In fact, such effective mixing give rise to high values of diffusion coefficients for mass, momentum and heat. The process is called Turbulent Diffusion. The largest turbulent eddies interact with and extract energy from the mean flow by a process called vortex stretching. In fact the presence of mean velocity gradients in sheared flows distort the rotational turbulent eddies; in particular, one end of the turbulent eddy is forced to move faster than the other. 4

5 Introduction The characteristic velocity and characteristic length of the larger eddies are of the same order of magnitude of the velocity scale and length scale of the mean flow. Hence a large eddies Reynolds number can be defined, that is of the same order of magnitude of the mean flow Reynolds number. Large eddies characteristic velocity Large eddies characteristic length Re Large Eddies = ρ ϑl µ Re = ρul µ Since the Re is very large for a turbulent flow, the large eddies are effectively inviscid and angular moment is conserved during vortex stretching. This means that the rotation speed increases as the vortex radius decreases and vice-versa. The stretching work done by the mean flow on the large eddies provides the necessary energy which maintains the turbulence. 5

6 Introduction Smaller eddies are themselves stretched strongly by larger eddies and more weakly by the mean flow. The kinetic energy is handed down from large eddies to progressively smaller and smaller eddies in the so-called energy cascade. The smallest scale of motion that can occur in a turbulent flow is dictated by viscosity and the Reynolds number of smallest eddies is equal to 1 In a typical engineering turbulent flow, the characteristic length of the smallest eddies is of the order of 0.1 to 0.01 mm, while the frequencies are around 10 MHz. Largest eddies Dominated by inertia forces Highly anisotropic (directional) structure. Smallest eddies Dominated by viscous forces Isotropic (diffusive) structure. 6

7 Turbulence modeling approaches In the study of turbulent flows the ultimate objective is to obtain a tractable quantitative theory of model that can be used to calculate quantities of interest and practical relevance. A century of experience has shown that the turbulence problem is vey difficult and that there are no prospects of a simple analytic theory. The hope is to use the even-increasing computing power to effectively describe turbulent flows. According to Bardina et al (1980) there are six categories of numerical methods for turbulent flows, most of which can be divided in sub-categories. 7

8 Turbulence modeling approaches 1 - USE OF CORRELATIONS The first method for turbulent flow calculations involve the use of correlations such as ones that give the friction factor as a function of the Reynolds number or the Nusselt number of heat transfer as a function of the Reynolds and Prandtl numbers. This method is limited to simple types of flows. Its application does not require the use of a computer. - USE OF INTEGRAL EQUATIONS The second method for turbulent flow calculations uses integral equation which can be derived from the equations of motion by integrating over one or more coordinates. Usually this reduces the problem to one or more ordinary differential equations which can be easily solved. 3 REYNOLDS AVERAGEG NAVIER STOKES (RANS) ONE POINT CLOSURE This method is obtained by averaging the equations of motion over time (if the flow is statistically steady) over a coordinate in which the mean flow does not vary. This approach is called One-Point- Closure and leads to a set of PDEs called Reynolds Averaged Navier Stokes (RANS). This equations do not form a closed set, so this method requires the introduction of approximations or turbulence models. 8

9 Turbulence modeling approaches 4 TWO POINT CLOSURE The fourth type of method is called two point closure. It uses equations for the correlation of velocity components at two spatial points or, more often, the Fourier transform of these equations. These methods are rarely used and will not be further discussed. 5 LARGE EDDY SIMULATION (LES) The Large Eddy Simulation (LES) method solves the largest scale of motion while approximates or models only the small scale of motions. It can be regarded as a kind of compromise between one point closure methods and direct numerical simulation approach. 6 DIRECT NUMERICAL SIMULATION (DNS) The Direct Numerical Simulation (DNS) method is based on the Navier Stokes equation resolution for all of the motion in a turbulent flow. 9

10 The RANS approach Engineers are normally interested in knowing just a few quantitative properties of a turbulent flow, such as the average forces on a body, the degree of mixing between two incoming streams of fluid, or the amount of substance that has reacted. Such a goal can be achieved using the RANS approach. In particular, all of the unsteadiness is averaged. In this way all the unsteadiness are considered as part of the turbulence. On averaging, the non-linearity of the Navier-Stokes equations are responsible for the presence of additional terms that must be modeled. EFFECT OF TURBULENCE ON TIME AVERAGED NAVIER-STOKES Let s firstly say that the mean quantity of a generic flow property can be defined as: Δ 1 t Φ = ϕ ( t) dt Δ t 0 10

11 The RANS approach In theory we should take the limit of time interval approaching to infinite, but Δt is large enough if it exceeds the time scales of the slowest variation of property ϕ (due to the largest eddies). This definition is adequate for steady mean flows. The flow property ϕ is time dependent and can be thought of as the sum of a steady mean component Φ and a time-varying fluctuating component ϕ, whit zero mean value. ϕ = Φ + ϕ' Δt 1 ϕ' = ϕ' ( t) dt 0 Δt 0 Information regarding the fluctuating part of the flow can, for example, be obtained from the Root- Mean-Square (RMS) of the fluctuations: Can be easily measured with for instance an hot wire-anenometer. Δt 1 ϕrms = ϕ = ϕ Δ t ( ') ( ') 0 dt 11

12 The RANS approach KINETIC ENERGY ASSOCIATED WITH THE TURBULENCE TURBULENT INTENSITY 1 k = u + v + w ( ) ' ' ' Reference mean flow velocity T i = k 3 U ref Kinetic energy RULES WHICH GOVERN THE TIME AVERAGES OF FLUCTUATING PROPERTIES (SCALAR) ϕ =Φ+ ϕ' ψ =Ψ+ ψ ' ϕ' = ψ ' = 0 ϕ =Φ ψ =Ψ ϕ+ ψ =Φ+Ψ ϕ ϕψ =ΦΨ+ ϕ ' ψ ' ϕds = Φ ds = s Φ s 1

13 The RANS approach RULES WHICH GOVERN THE TIME AVERAGES OF FLUCTUATING PROPERTIES (VECTORS AND SCALARS) Since divergence and gradient operators are both differentiations the rules the averages can be extended to a fluctuating vector a=a+a and its combination with a fluctuating scalar ϕ=φ+ϕ diva= diva ( ϕ ) = ( ϕ ) = ( ΦA) + ( ϕ' ') div( gradϕ ) = div( gradφ) div a div a div div a 13

14 The RANS approach INFLUENCE OF THE TURBULENT FLUCTUATIONS ON THE MEAN FLOW Let consider the instantaneous continuity and Navier-Stokes equations for an incompressible flow with constant viscosity. u = U + u' v= V + v' w= W + w' p = P+ p' u = 0 u 1 p + = + t ρ x ( uu) υ ( u) v 1 p + = + t ρ y ( vu) υ ( v) w 1 p + = + t ρ z ( wu) ν ( w) 14

15 The RANS approach INFLUENCE OF THE TURBULENT FLUCTUATIONS ON THE MEAN FLOW The time average is taken applying the rules shown in previous slides. u= U u U u' = + t t t ( uu) ( UU) ( u' u' ) = + 1 p 1 P = ρ x ρ x υ ( u) = ν ( U) U = 0 U t V t W t 1 P + + = + ρ x ( UU) ( u' u' ) υ ( U) 1 P + + = + ρ y ( VU) ( v' u' ) υ ( V) 1 P + + = + ρ z ( WU) ( w' u' ) υ ( W) I II III IV V 15

16 The RANS approach INFLUENCE OF THE TURBULENT FLUCTUATIONS ON THE MEAN FLOW The terms (I), (II), (IV) and (V) also appear in the instantaneous equations, but the time averaging process has introduced extra terms (III), which involve products of fluctuating velocities and constitute convective momentum transfer due to the velocity fluctuations. This terms are usually moved on the Right Hand Side (RHS) of the equations to reflect the role as additional turbulent stresses on mean velocity components. The turbulent flow equations for incompressible flows are (REYNOLDS EQUATIONS) U = 0 U P u u v u w + ( UU) = + υ ( U) + t ρ x x y z 1 ' ' ' ' ' V P v u v v w + ( VU) = + υ ( V) + t ρ y x y z 1 ' ' ' ' ' W 1 P w' u' w' v' w' + ( WU) = + υ ( W) + t ρ z x y z Six extra stresses termed Reynolds stresses. The normal stresses are always non-zero because they contains squared velocity fluctuations. On the contrary the shear stresses, if statistically independent, would be zero. However, the turbulent shear stresses are also non-zero and very large compared to the viscous stresses. 16

17 The RANS approach TRANSPORT EQUATION FOR AND ARBITRARY SCALAR QUANTITY The time averaged transport equation for the scalar j quantity is Φ u' ϕ' v' ϕ' w' ϕ' + ( Φ U) = ( ΓΦ Φ ) + + t x y z S Φ Closure problem ü The instantaneous continuity and Navier-Stokes equations form a closed set of four equations with four unknowns. ü In performing the time-averaging operation on the momentum equations we throw away all details concerning the state of the flow contained in the instantaneous fluctuations. As a result, we obtain six additional unknowns, the Reynolds stresses, in the time averaged flow equations. ü The complexity of turbulence usually precludes simple formulae for the extra terms. 17

18 The RANS approach: simple turbulent flows FREE TURBULENT FLOWS Free turbulent flows are among the simplest flows of significant engineering importance: mixing layers, jets and wakes. Jet Mixing Layer Wake 18

19 The RANS approach: simple turbulent flows FLAT PLATE BOUNDARY LAYER The presence of the solid boundary makes the flow behavior and turbulent structure considerably different from free turbulent flows. Dimensional analysis has greatly assisted in correlating experimental data. In turbulent thin shear layer flows the Reynolds number based on the length scale in the flow direction (L) is always very large, implying that the inertia forces are very large compared to viscous forces. 19

20 The RANS approach: simple turbulent flows If we form a Reynolds number based on a distance y away from the wall: Re y = Uy υ If the value of y is on the order of L, the inertia forces dominate in the flow As y is decreased to zero, the Re y also decreases to zero. As a consequence, there will be a range of values of y for which O(Re y ) 1. At this distance from the wall and closer the viscous forces will be of the same order of magnitude of inertia forces, or larger. To sum up, in flows along a solid walls, there is a inertia-dominated flow region far away from the wall and a thin layer within viscous effects are important. 0

21 The RANS approach: simple turbulent flows ü Close to the wall the flow is dominated by viscous forces and does not depend on free stream parameters. ü The mean flow velocity only depends on the distance y from the solid wall, fluid density, fluid viscosity and wall shear stress. Fluid density Mean flow velocity U = f ( y, ρµτ,, w ) Wall shear stress Wall distance Fluid viscosity 1

22 The RANS approach: simple turbulent flows The appropriate length and velocity scales are the distance form the wall, y, and the so-called friction velocity u t that is obtained form the dimensional analysis. τ u w τ = ρ Considering for sake of clearness just the averaged x-momentum equation: u + µ u =... + * x ρu τ y u * x 1/y+ Where: u U = and y = u + + τ ρuy τ µ Dimensionless wall distance

23 The RANS approach: simple turbulent flows Different layers can be individuated as a function of the y + value y + + y < 5 Linear sublayer + ρuy τ 5 y < 30 Buffer layer = + µ 30 y < 500 Log-law layer + y 500 Outer layer Linear Sub-layer At the solid surface the fluid is stationary, meaning that the turbulent eddies also stop very close to the wall. The fluid flow closest to the wall is then dominated by viscous shear. This layer is extremely thin (y + <5). It can be easily observed that in this layer: U τ µ τ y ( ) ( ) w y = w U y = y+ C U(0) = 0 C = 0 τ µ U τ w = y µ + U τw y + ρuτ y u τw ρuτ u uτ µ uτ µ uτ = = = = = y + 3

24 The RANS approach: simple turbulent flows Buffer layer In the buffer layer viscous and turbulent stresses are of similar magnitude. The linear sub-layer and the buffer layer form together the viscous sub-layer. Log-law layer Outside the viscous sub-layer a region exist where viscous and turbulent effects are both important. In this region the shear stress t is assumed to be constant and equal to the wall shear stress. The numerical relation between u + and y + in the log-law layer is: u = ln y + B= ln Ey K = 0.4; B= 5.5 or E = 9.8 K K ( ) ( ) The values of K and B are universal constants valid for all turbulent flows over smooth walls and at high Reynolds number. The logarithmic relationship is often called log-law and so the layer for 30<y + <500 is called lag-low layer. 4

25 The RANS approach: simple turbulent flows Outer layer From the experiments it has been observed that the log-law is valid in the region 0.0<y/d<0.. For larger values of y the correct velocity profile is given by the velocity defect law: Umax U 1 y = ln + u K δ τ A Where A is a constant. This law is often referred as the law of the wake. The experiments about the outer layer show that the turbulent layer adjacent composed basically by two region: to a solid wall I 1. The inner region, constituting 10 to 0% of the total thickness of the wall layer and composed by the linear sub-layer, the buffer layer and the log-law layer;. The outer region layer that an inertia dominated core flow and it is free from direct viscous effects 5

26 The RANS approach: simple turbulent flows In the figure it can be seen the very good agreement between theoretical relations for the viscous sublayer and the log-low layer and their areas of validity. 6

27 The RANS approach: turbulence models A turbulence model is a computational procedure to close the Reynolds equations. The most common turbulence models are: Classical Models Zero equation model mixing lenght model One-equation model (Spalart-Allmaras) Two-equation model (k-ε model) Reynolds stress equation model Algebraic stress model The classical models are based on the assumption that exists an analogy between viscous stresses and Reynolds stresses on the mean flow. It was proposed by Bousinnesq in 1877 that the Reynolds stresses can be related to mean rates of deformation: U U i j τij ρ ui ' uj ' µ = = t + xj x i Eddy viscosity (Pa s) 7

28 The RANS approach: turbulence models Turbulent transport of heat, mass and other scalar quantities is modeled in a similar way. Φ ρui' ϕ' =Γt x i Turbulent diffusivity Since turbulent transport of momentum and heat or mass transport are related to the same mechanism, that is the eddy mixing, it is expected that the value of the turbulent diffusivity is close to the value of the eddy viscosity. The ration between this two quantities is given by the Prandtl/Schmidt number, defined as follows: µ t σ t = Γ t Many experimental observations have shown that the Prandtl/Schmidt number is nearly constant and in most CFD procedure it is assumed to be equal to 1. 8

29 The RANS approach: mixing length model From the dimensional point of view, we assume that the cinematic viscosity can be expressed as a product of a turbulent velocity scale and a length scale: [ υ ] = [ ϑ] [ l] t Length scale (m) Cinematic viscosity (m /s) Turbulent vel. scale (m/s) If this is the case, then the effects of turbulence dimensional analysis yields: υ t = C1ϑ l Where C 1 is a dimensionless constant of proportionality. Most the kinetic energy is contained in the largest eddies and so the turbulent length scale is characteristic of this eddies that interact with the mean flow 9

30 The RANS approach: mixing length model Accepting the correlation between largest eddies and turbulent characteristic length, we can attempt to relate the characteristic velocity scale of the eddies with the mean flow properties. This has been observed to work well for simple D flows, for which the only significant gradient is U y In this condition, it is dimensionally correct to state that U ϑ = Cl y Where C is a dimensionless constant. Combining the found relation it is derived the Prandtl s mixing length model: υt = C1ϑ l U U U υt = CCl 1 = lm ϑ = Cl y y y 30

31 The RANS approach: mixing length model Mixing length for D turbulent flows (Versteeg and Malalasekera) 31

32 The RANS approach: k-e model The standard k-e model has two model equations, one for k, the turbulent kinetic energy, and one for ε, the rate of viscous dissipation, based on our best understanding of the relevant processes causing changes to these variables. The k and e variable are used to define velocity scale and length scale, representative of the large scale turbulence: ϑ = k l = 3 k ε Velocity scale Length scale The k-ε model includes a small eddy variable (ε) to define the large eddy scale because at high Reynolds numbers the rate at which large eddies extract energy from the mean flow is precisely matched to the rate of transfer of energy across the energy spectrum to small, dissipating, eddies. This aspects limits the energy growth or decrease at some scale of turbulence, in the model. Of course this does not occur in practice and justify the use of ε in the model. 3

33 The RANS approach: k-e model Applying the same approach as in the mixing length model: k µ t = ρυt = ρcϑl = ρc µ ε Transport equations used in the standard model ( ) k Uk i ν T k U U i j Ui + = ν + + νt + ε t xi xi σ k x i xj x i xj ( U ε ) ν U U j U ε i T ε ε i i ε + = ν + + Cε1 νt + Cε t xi xi σ ε xi k xj x i xj k C = 0.09 σ = 1 C = 1.44 C = 1.9 σ = 1.3 (Jones and Launder) µ k ε1 ε ε 33

34 The RANS approach: k-e model The meaning of different terms in the transport equations is: Rate of change of k or ε Transport of k or e by convection Transport of k or ε by diffusion Rate of production of k or ε Rate of destruction of k or ε To compute the Reynolds stresses with the k-ε model an extended Bousinnesq relationship is used: ρ U U i j u' i u' µ j = t + k ij xj x i 3 ρ δ The extra term makes the formula applicable to the normal Reynolds stresses. In any flow the sum of normal stresses is equal to minus twice the kinetic energy. The 1/3 is due to the fact that an equal third is allocated to each normal stress component to ensure their sum is always physically correct. 34

35 The RANS approach: k-e model The k-e equations are elliptic because of the gradient diffusion term. As a consequence the following boundary conditions are needed: Inlet Distribution od k and ε Outlet or symmetry Free streem k=0, ε=0 k n= 0 ε n= 0 Solid walls Approach depends on the Reynolds number In industrial CFD applications measurements of k and ε are rarely available. The inlet boundary conditions is typically imposed on the basis of data from the literature or observing the results sensitivity to the inlet distributions. If no information is available at all inlet distributions can be approximated form turbulent intensity T i and characteristic length of the domain, L k k = ( Uref Ti ) ; ε = Cµ ; l = 0.07L l 35

36 The RANS approach: k-e model Solid walls boundary conditions: HIGH REYNOLDS NUMBER At high Reynolds number the standard k-ε model makes use of the universal behavior of near wall flows discussed in the previous slides. In particular, the mean velocity at a point y p with 30<y p+ <500 satisfies the log-law and the rate of turbulent production equals the rate of dissipation. Solid walls boundary conditions: LOW REYNOLDS NUMBER At low Reynolds numbers the log-law is not valid so the above-mentioned boundary conditions cannot be used. As a consequence wall damping needs to be applied to ensure that viscous stresses take over from turbulent Reynolds stresses at low Reynolds numbers and the viscous sub-layer adjacent to solid walls. 36

37 The RANS approach: k-e model The k-ε model for the case of low Reynolds numbers is then modified as follows: ( ε ) ( ) µ = ρc f t k Uk i ν T k U U i j Ui + = ν + + νt + + νt EE ij ij ε t xi xi σ k x i xj x i xj ε Ui ν T ε ε U U i j Ui ε ε + = ν + + Cε1 νt + + C1 ε f1 νt EijEij Cε f t xi xi σ ε xi k xj x i xj k k µ µ k ε It can be seen that the diffusion terms include a viscous contribution and the constants are multiplied by a wall-dumping function. Example of wall damping functions are f µ R ( ) 0.5 k = 1 e 1+ Rt R k t f1 = 1+ f = 1 e where Rt = f µ νε 37

38 The RANS approach: turbulence models One equation models: Spalart-Allmaras In the Spalart-Allmaras (SA) model is a relatively new model. It is numerically forgiving, in terms of near wall resolution and stiffness, and yelds fairly rapid convergence to steady state. The SA model calculates the turbulent viscosity via an intermediate variable trough the relation: X ˆ ν ν = ˆ ν f = ˆ ν where X = and f = 3 3 T v1 3 3 v1 3 3 X + cv 1 ν X + cv 1 The intermediate variable is computed through the following transport equation: X ( ) ˆν t + U ˆν j x j TRANSPORT = c b1 Ŝ ˆν + 1 ( ν + ˆν ) ˆν + c b ˆν ˆν σ x i x i σ x c ω1 f ω i y PRODUCTION TERM DIFFUSION TERM DESTRUCTION TERM Different constant values are given in the scientific literature. 38

39 Turbulent incompressible flow over a backward facing step 39

40 Turbulent incompressible flow over a backward facing step Results of the simulations performed using the non commercial AC-CBS and the SA turbulence model. The simulations have been performed in correspondence of a Reynolds number of 305, in accordance to M.K. Denham et al., Journal of Physics E: Scientific Instruments 1975 Volume 8. x- velocity contours and streamlines for a Reynolds number Re=305 Turbulence variable field obtained for a Reynolds number Re=305 40

41 3 Turbulent incompressible flow over a backward facing step COMPARISON OF THE OBTAINED VELOCITY PROFILES TO REFERENCE DATA FROM DENHAM ET AL. 3 3 y-axis y-axis y-axis AC-CBS x=1 Denham et al. x= x-velocity AC-CBS x= Denham at al. x= x-velocity AC-CBS x=4 Denham et al. x= x-velocity 41