2. Conservation Equations for Turbulent Flows

Transcription

1 2. Conservation Equations for Turbulent Flows Coverage of this section: Review of Tensor Notation Review of Navier-Stokes Equations for Incompressible and Compressible Flows Reynolds & Favre Averaging and RANS & FANS Equations Turbulent Kinetic Energy and Reynolds Stresses Closure Problem and Turbulence Modelling Review of Tensor Notation Tensor notation is used extensively throughout the textbook and this course and is therefore briefly reviewed and compared to vector notation before moving to a discussion of the conservation equations for turbulent flows. 2

2 2.1 Review of Tensor Notation Expression Vector Notation Tensor Notation scalars π, c π, c operations (zeroth-order tensor) (+,,, /) e.g., π c, π c π c, π c vectors a, x a i, x i (3D space) (first-order tensor, it is taken that i {1, 2, 3}) addition b = a + x bi = a i + x i = a j + x k vector products inner product a x = i a ix i = c a i x i = c (scalar result) i a i x i = a 1 x 1 + a 2 x 2 + a 3 x 3 3 (Einstein notation: sum implied) 2.1 Review of Tensor Notation Einstein Summation Convention Einstein summation convention: repetition of an index in any term denotes a summation of the term with respect to that index over the full range of the index (i.e., 1, 2, 3). Thus, for the inner product a i x i = 3 a i x i = a 1 x 1 + a 2 x 2 + a 3 x 3 i=1 the sum is implied and need not be explicitly expressed. Note that using matrix-vector mathematical notation, the inner product of two 3 1 column vectors, a and x, can be experssed as a T x = [a 1 a 2 a 3 ] x 1 x 2 x 3 = a 1 x 1 + a 2 x 2 + a 3 x 3 4

3 2.1 Review of Tensor Notation Expression Vector Notation Tensor Notation cross product a x = r = i j k a 1 a 2 a 3 x 1 x 2 x 3 ɛ ijk a j x k = r i (vector result) r = (a 2x 3 a 3 x 2 ) i (a 1 x 3 a 3 x 1 ) j +(a 1 x 2 a 2 x 1 ) k ɛ ijk = permutation tensor (sum over j & k implied) outer product a x = a x = J aα x β = J αβ (dyadic result, (second-order tensor, vector of vectors) 9 elements, 6 elements for symmetric tensor) Review of Tensor Notation Dyadic Quantity: A Vector of Vectors In vector notation, a dyadic quantity, d is essentially a vector of vectors as defined by the outer product: d = u v It is equivalent to the second-order tensor, d ij, d ij = u i u j using tensor notation. In this case using matrix-vector notation, the outer product of two 3 1 column vectors, u and v, can be experssed as uv T = u 1 u 2 u 3 [v 1 v 2 v 3 ] = 6 u 1 v 1 u 1 v 2 u 1 v 3 u 2 v 1 u 2 v 2 u 2 v 3 u 3 v 1 u 3 v 2 u 3 v 3

4 2.1 Review of Tensor Notation Expression Vector Notation Tensor Notation dyads d = u v dij = u i u j dyad-vector products (vector result) A x = b A αβ x β = b α equivalent to Ax = b high-order tensors Q Q ijk (third-order tensor, 27 elements, 10 symmetric) R 7 R ijkl (fourth-order tensor, 81 elements, 15 symmetric) 2.1 Review of Tensor Notation Expression Vector Notation Tensor Notation contracted quantities h hi = q ijj (contacted 3rd-order tensor, vector) P P ij = R ijkk p (contacted 4th-order tensor, second-order tensor, dyad) p = R iikk (double contacted tensor, scalar quantity) 8

5 2.1 Review of Tensor Notation Permutation Tensor The permuation tensor, ɛ ijk, is a third-order tensor that is introduced for defining cross products with the following properties for its elements: ɛ 123 = ɛ 231 = ɛ 312 = 1, even permutations ɛ 213 = ɛ 321 = ɛ 132 = 1, odd permutations ɛ 111 = ɛ 222 = ɛ 333 = 0, repeated indices ɛ 112 = ɛ 113 = ɛ 221 = ɛ 223 = ɛ 331 = ɛ 322 = 0, repeated indices Review of Tensor Notation Kronecker Delta Tensor The Kronecker delta tensor, δ ij, is a second-order tensor that is defined as follows: δ ij = { 1, for i = j 0, for i j The Kronecker delta tensor is equivalent ot the identity dyad, I and the 3 3 indentity matrix, I, in matrix-vector mathematical notation given by I = Note also that δ ii = trace(i) = 3 10

6 2.1 Review of Tensor Notation ɛ δ Indentity The following identity relates the permutation and Kronecker delta tensors: ɛ ijk ɛ ist = δ js δ kt δ jt δ ks Review of Tensor Notation Expression Vector Notation Tensor Notation differential operators gradient V = φ Vi = φ divergence c = a c = a i u φ u i φ curl g = a k a g i = ɛ ijk P = B vector derivative P ij = B i Laplacian c = 2 φ = φ c = 2 φ a = 2 A = A ai = 2 A i 12

7 2.1 Review of Tensor Notation Other Notation In the course textbook and elsewhere you will some time see the use of the shorthand tensor notation: and p = p = p,i u = u i = u i,i This notation will not be used by this instructor as it can be difficult to follow and is more prone to errors Navier-Stokes Equations for a Compressible Gas The Navier-Stokes equations describing the flow of compressible gases are a non-linear set of partial-differential equations (PDEs) governing the conservation of mass, momentum, and energy of the gaseous motion. They consist of two scalar equations and one vector equation for five unknowns (dependent variables) in terms of four independent variables, the three-component position vector, x or x i, and the scalar time, t. We will here review briefly the Navier-Stokes equations for a polytropic (calorically perfect) gas in both tensor and vector notation. Integral forms of the equations will also be discussed. 14

8 2.2 Navier-Stokes Equations for a Compressible Gas Continuity Equation The continuity equation is a scaler equation reflecting the conservation of mass for a moving fluid. Using vector notation, it has the form ρ t + (ρ u) = 0 where ρ and u are the gas density and flow velocity, respectively. In tensor notation, the continuity equation can be written as ρ t + (ρu i ) = Continuity Equation For the control volume and control surface above, the integral form of the continuity equation can be obtained by integrating the original PDE over the control volume and making using of the divergence theorem. The following integral equation is obtained: d dt V ρ dv = A ρ u n da which relates the time rate of change of the total mass within the control volume to the mass flux through the control surface. 16

9 2.2 Navier-Stokes Equations for a Compressible Gas Momentum Equation The momentum equation is a vector equation that represents the application of Newton s 2nd Law of Motion to the motion of a gas. It relates the time rate of change of the gas momentum to the forces which act on the gas. Using vector notation, it has the form t (ρ u) + ( ρ u u + p I τ ) = ρ f where p and τ are the gas pressure and fluid stress dyad or tensor, respectively, and f is the acceleration of the gas due to body forces (i.e., gravitation, electro-magnetic forces). In tensor notation, the momentum equation can be written as t (ρu i) + (ρu i u j + pδ ij τ ij ) = ρf i Momentum Equation For the control volume, the integral form of the momentum equation is given by ( ) d ρ u dv = ρ u u + p dt I τ n da + ρ f dv V A which relates the time rate of change of the total momentum within the control volume to the surface and body forces that act on the gas. V 18

10 2.2 Navier-Stokes Equations for a Compressible Gas Energy Equation The energy equation is a scalar equation that represents the application of the 1st Law of Thermodynamics to the gaseous motion. It describes the time rate of change of the total energy of the gas (the sum of kinetic energy of bulk motion and internal kinetic or thermal energy). Using vector notation, it has the form t (ρe) + [ ρ u ( E + p ρ ) ] τ u + q = ρ f u where E is the total specific energy of the gas given by E =e + u u/2 and q is the heat flux vector representing the flux of heat out of the gas. In tensor notation, it has the form t (ρe) + ( [ρu i E + p ) ] τ ij u j + q i ρ = ρf i u i Energy Equation For the control volume, the integral form of the energy equation is given by [ ( d ρe dv = ρ u E + p ) ] τ u + q n da+ ρ dt ρ f u dv V A V which relates the time rate of change of the total energy within the control volume to transport of energy, heat transfer, and work done by the gas. 20

11 2.2 Navier-Stokes Equations for a Compressible Gas The Navier-Stokes equations as given above are not complete (closed). Additional information is required to relate pressure, density, temperature, and energy, and the fluid stress tensor, τ ij and heat flux vector, q i must be specified. The equation set is completed by thermodynamic relationships; constitutive relations; and expressions for transport coefficients. When seeking solutions of the Navier-Stokes equations for either steady-state boundary value problems or unsteady initial boundary value problems, boundary conditions will also be required to complete the mathematical description Navier-Stokes Equations for a Compressible Gas Thermodynamic Relationships In this course, we will assume that the gas satisfies the ideal gas equation of state relating ρ, p, and T, given by p = ρrt and behaves as a calorically perfect gas (polytropic gas) with constant specific heats, c v and c p, and specific heat ratio, γ, such that e = c v T = p (γ 1)ρ and h = e + p ρ = c pt = γp (γ 1)ρ where R is the gas constant, c v is the specific heat at constant volume, c p is the specific heat at constant pressure, and γ =c p /c v. 22

12 2.2 Navier-Stokes Equations for a Compressible Gas Mach Number and Sound Speed For a polytropic gas, the sound speed, a, can be determined using a = γ p ρ = γrt and thus the flow Mach number, M, is given by M = u a = u γrt Navier-Stokes Equations for a Compressible Gas Constitutive Relationships The constitutive relations provide expressions for the fluid stress tensor, τ ij, and heat flux vector, q i, in terms of the other fluid quantities. Using the Navier-Stokes relation, the fluid stress tensor can be related to the fluid strain rate and given by τ ij = µ [( ui + u j ) 2 3 δ ij u k x k ] (τ ii = 0, traceless) where µ is the dynamic viscosity of the gas. Fourier s Law can be used to relate the heat flux to the temperature gradient as follows: q i = κ T or q = κ T where κ is the coefficient of thermal conductivity for the gas. 24

13 2.2 Navier-Stokes Equations for a Compressible Gas Transport Coefficients In general, the transport coefficients, µ and κ, are functions of both pressure and temperature: µ = µ(p, T ) and κ = κ(p, T ) Expressions, such as Sutherland s Law can be used to determine the dynamics viscosity as a function of temperature (i.e., µ=µ(t )). The Prandtl number can also be used to relate µ and κ. The non-dimensional Prandtl number is defined as follows: Pr = µc p κ and is typically for many gases. Given µ, the thermal conductivity can be related to viscosity using the preceding expression for the Prandtl number Navier-Stokes Equations for a Compressible Gas Boundary Conditions At a solid wall or bounday, the following boundary conditions for the flow velocity and temperature are appropriate: u = 0, (No-Slip Boundary Condition) and T = T wall, (Fixed Temperature Wall Boundary Condition) or T n = 0, (Adiabatic Wall Boundary Condition) where T wall is the wall temperature and n is a unit vector in the direction normal to the wall or solid surface. 26

14 2.3 Navier-Stokes Equations for an Incompressible Gas For low flow Mach numbers (i.e., low subsonic flow, M<1/4), the assumption that the gas behaves as an incompressible fluid is generally a good approximation. By assuming that the density, ρ, is constant; temperature variations are small and unimportant such that the energy equation can be neglected; and the viscosity, µ, is constant; one can arrive at the Navier-Stokes equations describing the flow of incompressible fluids Navier-Stokes Equations for an Incompressible Gas Continuity Equation Using vector notation, the continuity equation for incompressible flow reduces to u = 0 In other words, the velocity vector, u, is a solenoidal vector field and is divergence free. In tensor notation, the solenoidal condition can be expressed as u i = 0 28

15 2.3 Navier-Stokes Equations for an Incompressible Gas Momentum Equation Using vector notation, the momentum equation for an incompressible fluid can be written as u t + u u + 1 ρ p = 1 ρ τ In tensor notation, the incompressible form of the momentum equation is given by u i t + u u i j + 1 p = 1 τ ij ρ ρ Navier-Stokes Equations for an Incompressible Gas Constitutive Relationships For incompressible flows, the Navier-Stokes constitutive relation relating the fluid stresses and fluid strain rate can be written as ( ui τ ij = µ + u ) ( j ui = ρν + u ) j = 2ρνS ij where ν =µ/ρ is the kinematic viscosity and the strain rate tensor (dyadic quantity) is given by S ij = 1 ( ui + u ) j 2 As in the compressible case, the fluid stress tensor for incompressible flow is still traceless and τ jj =0. 30

16 2.3 Navier-Stokes Equations for an Incompressible Gas Vorticity Transport Equation The vorticity vector, Ω, is related to the rotation of a fluid element and is defined as follows: Ω = u or Ω i = ɛ ijk u k For incompressible flows, the momentum equation can be used to arrive at a transport equation for the flow vorticity given by Ω t u Ω = ν 2 Ω Vorticity Transport Equation Using u Ω = Ω u u Ω, the vorticity transport equation can be re-expressed as Ω t + u Ω Ω u = ν 2 Ω Using tensor notation, this equation can be written as Ω i t + u Ω i j Ω j u i = ν 2 Ω i 32

17 2.4 Reynolds Averaging As discussed previously, turbulent flow is characterized by irregular, chaotic motion. The common approach to the modelling of turbulence is to assume that the motion is random and adopt a statistical treatment. Reynolds (1895) introduced the idea that the turbulent flow velocity vector, u i, can be decomposed and represented as a fluctuation, u i, about a mean component, U i, as follows: u i = U i + u i One can then develop and solve conservation equations for the mean quantities (i.e., the Reynolds-averaged Navier-Stokes (RANS) equations) and incorporate the influence of the fluctuations on the mean flow via turbulence modelling Reynolds Averaging Forms of Reynolds Averaging 1. Time Averaging: appropriate for steady mean flows 1 t+t /2 F T ( x) = lim f ( x, t ) dt T T t T /2 2. Spatial Averaging: suitable for homogeneous turbulent flows 1 F V (t) = lim f ( x, t) dv V V 3. Ensemble Averaging: most general form of averaging 1 N F E ( x, t) = lim f n ( x, t) N N n=1 where f n ( x, t) is nth instance of flow solution with initial and boundary data differing by random infinitessimal perturbations. V 34

18 2.4.1 Forms of Reynolds Averaging For ergodic random processes, these three forms of Reynolds averaging will yield the same averages. This would be the case for stationary, homogeneous, turbulent flows. In this course and indeed in most turbulence modelling approaches, time averaging will be considered. Note that Wilcox (2002) states that Reynolds time averaging is a brutal simplification that loses much of the information contained in the turbulence Reynolds Averaging Reynolds Time Averaging In Reynolds time averaging, all instantaneous flow quantities, φ(x i, t) and a(x i, t), will be represented as a sum of mean and fluctuating components, Φ(x i ) and φ (x i, t) and A(x i ) and a (x i, t), respectively, such that φ(x i, t) = Φ(x i ) + φ (x i, t) or a(x i, t) = A(x i ) + a (x i, t) For the flow velocity, we have u i (x α, t) = U i (x α ) + u i(x α, t) 36

19 2.4.2 Reynolds Time Averaging The time averaging procedure is defined as follows and yields the time averaged quantities: φ(x i, t) = Φ(x i ) = lim T 1 T t+t /2 t T /2 φ(x i, t ) dt a(x i, t) = A(x i ) = lim T 1 T t+t /2 t T /2 a(x i, t ) dt By definition, time averaging of mean quantities merely recovers the mean quantity: U i (x α ) = lim T 1 T t+t /2 t T /2 U i (x α ) dt = U i (x α ) Reynolds Time Averaging Similarly by definition, time averaging of time-averaged quantities yields zero: u i (x α, t) = lim T 1 T t+t /2 t T /2 [ ui (x α, t ) U i (x α ) ] dt = 0 38

20 2.4 Reynolds Averaging Separation of Time Scales In practice, the time period for the averaging, T, is not infinite but very long relative to the time scales for the turbulent fluctuations, T 1 ( i.e., T T 1 ). This definition of time averaging and T works well for stationary (steady) flows. However, for non-stationary (unsteady flows), the validity of the Reynolds time averaging procedure requires a strong separation to time scales with T 1 T T 2 where T 2 is the time scale for the variation of the mean Reynolds Averaging Separation of Time Scales T 1 u(x,t) T 2 t T 1 T T 2 40

21 2.4.3 Separation of Time Scales Provided there exists this separation of scales, the time averaging procedure for time-varying mean flows can be defined as follows: with T 1 T T 2. φ(x i, t) = Φ(x i, t) = 1 T a(x i, t) = A(x i, t) = 1 T t+t /2 t T /2 t+t /2 t T /2 φ(x i, t ) dt a(x i, t ) dt Reynolds Averaging Properties of Reynolds Time Averaging Multiplication by a scalar: c a(x i, t) = c T t+t /2 t T /2 a(x i, t ) dt = ca Spatial differentiation: a = 1 T t+t /2 t T /2 a dt = ( 1 T t+t /2 t T /2 a dt ) = A 42

22 2.4 Reynolds Averaging Properties of Reynolds Time Averaging Temporal differentiation: u i t = 1 T t+t /2 t T /2 u i t dt = U i(x i, t + T /2) U i (x i, t T /2) T U i t + u i (x i, t + T /2) u i (x i, t T /2) T The latter is obtained by assuming that u U and T T Reynolds Averaging Single-Point Correlations What about time-averaged products? a(x i, t)b(x i, t) = (A + a ) (B + b ) = AB + a B + b A + a b = AB + Ba + Ab + a b = AB + Ba + Ab + a b = AB + a b In general, a and b are said to be correlated if a b 0 and uncorrelated if a b = 0 44

23 2.4 Reynolds Averaging Single-Point Correlations What about triple products? Can show that a(x i, t)b(x i, t)c(x i, t) = ABC + a b C + a c B + b c A + a b c Reynolds Averaged Navier-Stokes (RANS) Equations Derivation Applying Reynolds time-averaging to the incompressible form of the Navier-Stokes equations leads to the Reynolds Averaged Navier-Stokes (RANS) equations describing the time variation of mean flow quantities. Application of time-averaging to the continuity equations yields or u i = 0 U i = 0 46

24 2.5 Reynolds Averaged Navier-Stokes (RANS) Equations Derivation For the incompressible form of the momentum equation we have u i t + u u i j + 1 ρ p = u i t + u u i j + 1 ρ p = 1 τ ij ρ Considering each term in the time-averaged equation above we have: u i t = U i t 1 ρ p = 1 ρ p = 1 P ρ Reynolds Averaged Navier-Stokes (RANS) Equations Derivation 1 τ ij ρ = 1 ρ τ ij = 2 ρ ρν S ij where the mean strain, S ij, is defined as S ij = 1 2 [ Ui + U j = 2ν S ij ] 48

25 2.5 Reynolds Averaged Navier-Stokes (RANS) Equations Derivation u i u j = u j (u i u j ) u i = ) (U i U j + u i x u j j = (U i U j ) + ) (u i x u j j U i U j = U j + U i + ) (u i x u j j U i = U j + ) (u i x u j j Thus we have U i t + U U i j + 1 ρ P = 1 ) (2µ S ij ρu i ρ x u j j Reynolds Averaged Navier-Stokes (RANS) Equations Summary In summary, the RANS describing the time-evolution of the mean flow quantities U i and P can be written as U i t + U U i j + 1 ρ U i = 0 P = 1 ( τ ij + λ ij ) ρ where τ ij is the fluid stress tensor evaluated in terms of the mean flow quantities and λ ij is the Reynolds or turbulent stress tensor given by λ ij = ρu i u j 50

26 2.6 Reynolds Turbulent Stresses and Closure Problem Closure or RANS Equations The Reynolds stresses λ ij = ρu i u j incorporate the effects of the unresolved turbulent fluctuations (i.e., unresolved by the mean flow equations and description) on the mean flow. These apparent turbulent stresses significantly enhance momentum transport in the mean flow. The Reynolds stress tensor, λ ij, is a symmetric tensor incorporating six (6) unknown or unspecified values. This leads to a closure problem for the RANS equation set. Turbulence modelling provides the necessary closure by allowing a means for specifying λ ij in terms of mean flow solution quantities Reynolds Turbulent Stresses and Closure Problem Reynolds Stress Transport Equations Transport equations for the Reynolds stresses, λ ij = ρu i u j can be derived by making use of the original and time-averaged forms of the momentum equations. Starting with the momentum equation for incompressible flow governing the time evolution of the instantaneous velocity vector, u i, and noting that 1 τ ij ρ = µ ρ u i t + u u i j + 1 ρ ( ui + u j ) = ν p = 1 τ ij ρ [ 2 u i + ( )] uj = ν 2 u i 52

27 2.6.2 Reynolds Stress Transport Equations one can write Similarily, Thus, u j (1) + u i u i t + u u i k + 1 x k ρ u j t + u u j k + 1 x k ρ ( 0 = u j ui t + u k +u i p ν p ν (2) can be written as u i x k + 1 ρ ( uj t + u u j k + 1 x k ρ 2 u i x k x k = 0 (1) 2 u j x k x k = 0 (2) p ν p ν 2 u i x k x k 2 u j ) x k x k ) Reynolds Stress Transport Equations The various terms appearing in the preceding equation can be expressed as follows: u j u i t + u j u i t = u j t ( Ui + u i) + u i t ( ) U j + u j = U i t u j + u u i j t + U j t u i + u i = u j u i t + u i = ( ) u i t u j = 1 λ ij ρ t u j t u j t 54

28 2.6.2 Reynolds Stress Transport Equations u j ρ p + u i ρ p = u j ρ (P + p x ) + u i i ρ = P u j x + 1 i ρ u j = 1 [ u p j + u i ρ p + P p (P + p ) u i x + 1 j ρ u i ] p Reynolds Stress Transport Equations νu j 2 u i x k x k + νu i 2 u j x k x k = νu j 2 x k x k (U i + u i ) + νu i 2 ( ) Uj + u j x k x k = ν 2 U i u j + νu 2 u i j + ν 2 U j 2 u u i + νu j i x k x k x k x k x k x k x k x k = νu j 2 u i 2 u + νu j i x k x k x k x k 2 ) = ν (u i x k x u j 2ν u u i j k x k x k = ν ρ 2 λ ij x k x k 2ν u i x k u j x k 56

29 2.6.2 Reynolds Stress Transport Equations u j u u i k + u i x u u j k = u j k x (U k + u k ) (U i + u i k x ) + u i (U k + u k ) ( ) Uj + u j k x k ) = U k (u i x u j + u U i j u k + u U j i k x u k k x k U i +U k u U j j + U k u i + u k x k x k λ ij λ jk U i x k ρ x k + ) (u i x u j u k u i u j k λ ij λ jk U i x k ρ x k = U k ρ = U k ρ λ ik ρ u k x k λ ik ρ ( ) u u j k U j x k U j x k + ) (u i x u j u k k Reynolds Stress Transport Equations Combining all of these terms, can write λ ij t + U k λ ij x k + λ jk U i x k + λ ik U j x k = x k +u j [ ν λ ] ij + ρu i x u j u k k p + u p i +2µ u i x k u j x k The preceding is a transport equation describing the time evolution of the Reynolds stresses, λ ij. 58

30 2.6.2 Reynolds Stress Transport Equations While providing a description for the transport of λ ij, the Reynolds stress equations introduce a number of other correlations of fluctuating quantities: u j p ρu i u j u k : symmetric second-order tensor, 6 entries : symmetric third-order tensor, 10 entries 2µ u i x k u j x k : symmetric second-order tensor, 6 entries leading to 22 additional unknown quantities. This illustrates well the closure problem for the RANS equations Turbulence Intensity and Kinetic Energy Turbulent Kinetic Energy Turbulent kinetic energy contained in the near-randomly fluctuating velocity of the turbulent motion is important in characterizing the turbulence. The turbulent kinetic energy, k, can be defined as follows: k = 1 2 u i u i = 1 2 (u 2 + v 2 + w 2 ) = 1 2 λ ii ρ = 1 2ρ (λ xx + λ yy + λ zz ) where u 2 = λ xx /ρ, v 2 = λ yy /ρ, and w 2 = λ zz /ρ. 60

31 2.7 Turbulence Intensity and Kinetic Energy Turbulence Intensity Relative turbulence intensities can be defined as follows: u û = 2 v, ˆv = 2 w, ŵ = 2 U U where U is a reference velocity. For isotropic turbulence, u 2 = v 2 =w 2, and thus 2 k û = ˆv = ŵ = 3 For flat plate incompressible boundary layer flow, U =U, û >0.10, and the turbulence is anisotropic such that U 2 u 2 : v 2 : w 2 = 4 : 2 : 3 61 U Turbulence Intensity 62

32 2.8 Turbulent Kinetic Energy Transport Equation Derivation Can derive a transport equation for the turbulent kinetic energy through contraction of the Reynolds stress transport equations using the relation that k = 1 2 u i u i = 1 2 The following equation for the transport of k can be obtained: k t +U i k = λ ij ρ U i + λ ii ρ ( ν k 1 ρ p u i 1 ) 2 u i u k u k ν u i u i As for the Reynolds stress equations, a number of unknown higher-order correlations appear in the equation for k requiring closure Turbulent Kinetic Energy Transport Equation Discussion of Terms Terms in this transport equation can be identified as follows: Production: U i k k t : time evolution of k : convection transport of k λ ij ρ U i : production of k by mean flow 64

33 2.8.2 Discussion of Terms Diffusion: ν k 1 ρ p u i 1 2 u i u k u k : molecular diffusion of k : pressure diffusion of k : turbulent transport of k Discussion of Terms Dissipation: ν u i u i = ɛ : dissipation of k at small scales where ɛ is the dissipation rate of turbulent kinetic energy. 66

34 2.9 Kinetic Energy Spectrum and Kolmogorov -5/3 Law Definition Further insight into the energy contained in the unresolved turbulent motion can be gained by considering the turbulent kinetic energy spectrum. The turbulent kinetic energy can be expressed as k = 0 E(κ)dκ where E(κ) is the spectral distribution of turbulent energy, κ is the wave number of the Fourier-like energy mode, and l is the wave length of the energy mode such that E(κ)dκ = turbulent energy contained between κ and κ + dκ and where l = 1 κ Kinetic Energy Spectrum and Kolmogorov -5/3 Law 68

35 2.9 Kinetic Energy Spectrum and Kolmogorov -5/3 Law Slope 2 Slope -5/3 E(κ) Energy-containing range Inertial subrange Dissipation range κ EI κ κ DI η Kinetic Energy Spectrum and Kolmogorov -5/3 Law Range of Turbulent Scales The large-scale turbulent motion (κ 0) contains most of the turbulent kinetic energy, while most of the vorticity resides in the small-scale turbulent motion (κ 1/η), where η, the Kolmogorov scale, is the smallest scale present in the turbulence. The dissipation of the turbulence kinetic energy occurs at the Kolmogorov scale and it follows from Kolmogorov s universal equilibrium theory and his first similarity hypothesis that dk dt = ɛ, and η = ( ν 3 ɛ ) 1/4 70

36 2.9.2 Range of Turbulent Scales For high Reynolds number turbulence, dimensional analysis and experimental measurements confirm that the dissipation rate, ɛ, turbulent kinetic energy, k, and largest scale representing the large scale motions (i.e., scale of the largest eddies), l 0, are related as follows: ɛ k3/2 l 0 When discussing features of turbulence, it was noted that it contains a wide range of scales. This implies that l 0 η Range of Turbulent Scales Using the expression above for l 0, an examination of the length scales reveals that ( ) 1/4 ( ) 3/4 l 0 η = l 0 (ν 3 /ɛ) 1/4 l 0 k 3/2 k 1/2 l 0 ν 3/4 Re 3/4 t ν l 0 where Re t is the turbulent Reynolds number. Thus l 0 η for high turbulent Reynolds number flows (i.e., for Re t 1). The latter is a key assumption entering into Kolmogorov s universal equilibrium theory and his three hypotheses. 72

37 2.9 Kinetic Energy Spectrum and Kolmogorov -5/3 Law Kolmogorov -5/3 Law Kolmogorov also hypothesized an intermediate range of turbulent scales lying between the largest scales and smallest scales where inertial effects dominate (this is the basis for the second similarity hypothesis). He postulated that in this inertial sub-range, E(κ) only depends on κ and ɛ. Using dimensional analysis he argued that E(κ) = C k ɛ 2/3 κ 5/3 or E(κ) κ 5/ Kinetic Energy Spectrum and Kolmogorov -5/3 Law 74

38 2.9.3 Kolmogorov -5/3 Law Although the Kolmogorov -5/3 Law is not of prime importance to RANS-based turbulence models, it is of central importance to DNS and LES calculations. Such simulations should be regarded with skeptism if they fail to reproduce this result Two-Point Correlations Two-Point Velocity Correlations So far we have only considered single-point or one-point correlations of fluctuating quantities. Two-point correlations are useful for characterizing turbulence and, in particular, the spatial and temporal scales and non-local behaviour. They provide formal definitions of the integral length and time scales characterizing the large scale turbulent motions. There are two forms of two-point correlations: two-point correletions in time; and two-point correlations in space. Both forms are based on Reynolds time averaging. 76

39 Two-Point Velocity Correlations Two-Point Autocorrelation Tensor (In Time): R ij (x i, t; t ) = u i (x i, t)u j (x i, t + t ) Two-Point Velocity Correlation Tensor (In Space): R ij (x i, t; r i ) = u i (x i, t)u j (x i + r i, t) For both correlations, k(x i, t) = 1 2 R ii(x i, t; 0) Two-Point Correlations Integral Length and Time Scales The integral length and time scales, τ and l, can be defined as follows: l(x i, t) = 3 R ii (x i, t; r) dr 16 k(x i, t) τ(x i, t) = 0 0 R ii (x i, t; t ) 2k(x i, t) dt where r = r i = r i r i and 3/16 is a scaling factor. 78

40 2.10 Two-Point Correlations Taylor s Hypothesis The two types of two-point correlations can be related by applying Taylor s hypothesis which assumes that t = U i This relationship assumes that u i U i and predicts that the turbulence essentially passes through points in space as a whole, transported by the mean flow (i.e., assumption of frozen turbulence) Favre Time Averaging Reynolds Time Averaging for Compressible Flows If Reynolds time averaging is applied to the compressible form of the Navier-Stokes equations, some difficulties arise. In particular, the original form of the equations is significantly altered. To see this, consider Reynolds averaging applied to the continuity equation for compressible flow. Application of time-averaging to the continuity equations yields t ρ t + (ρu i ) = 0 ( ρ ) + ρ + [( ρ + ρ x ) ( ) ] U i + u i = 0 i 80

41 Reynolds Time Averaging for Compressible Flows The Reynolds time averaging yields t ( ρ) + ] [ ρu i + ρ x u i = 0 i The introduction of high-order correlations involving the density fluctuations, such as ρ u i, can complicate the turbulence modelling and closure. Some of the complications can be circumvented by introducing an alternative time averaging procedure: Favre time averaging, which is a mass weighted time averaging procedure Favre Time Averaging Definition Favre time averaging can be defined as follows. The instantaneous solution variable, φ, is decomposed into a mean quantity, φ, and fluctuating component, φ, as follows: The Favre time-averaging is then φ = φ + φ ρφ(x i, t) = 1 T t+t /2 t T /2 ρ(x i, t )φ(x i, t ) dt = ρ φ + ρφ = ρ φ where φ(x i, t) 1 ρt t+t /2 t T /2 ρ(x i, t )φ(x i, t ) dt, ρφ 0 82

42 2.11 Favre Time Averaging Comparison of Reynolds and Favre Averaging Decomposition Time Averaging Reynolds : φ = φ + φ, Favre : φ = φ + φ Reynolds : φ = φ + φ = φ, Favre : ρφ = ρ( φ + φ ) = ρ φ Fluctuations Reynolds : φ = 0, Favre : ρφ = Comparison of Reynolds and Favre Averaging Further comparisons are possible. For Reynolds averaging we have ρφ = ρ φ + ρ φ and for Favre averaging we have ρφ = ρ φ Thus or ρ φ = ρ φ + ρ φ φ = φ + ρ φ ρ 84

43 Comparison of Reynolds and Favre Averaging We also note that To see this, start with φ 0 φ = φ φ = φ φ ρ φ Now applying time averaging, we have φ = φ φ ρ φ ρ = φ φ ρ φ ρ = φ ρ ρ 0 ρ Comparison of Reynolds and Favre Averaging Returning to the compressible form of the continuity equation, we can write ρu i = ρu i + ρ u i = ρũ i and therefore the Favre-averaged form of the continuity equation is given by t ( ρ) + ( ρũ i ) = 0 It is quite evident that the Favre-averaging procedure has recovered the original form of the continuity equation without introducing additional high-order correlations. 86

44 2.12 Favre-Averaged Navier-Stokes (FANS) Equations Continuity Equation: Momentum Equation: t ( ρ) + ( ρũ i ) = 0 t ( ρũ i) + ( ρũ i ũ j + pδ ij ) = ( ) τ ij ρu i x u j j Favre-Averaged Reynolds Stress Tensor: Turbulent Kinetic Energy: 1 2 ρu i u i λ = ρu i u j = 1 2 λ ii = ρ k Favre-Averaged Navier-Stokes (FANS) Equations Energy Equation: [ ρ (ẽ + 12ũiũ ) i t ρu i u i = [( + ] + τ ij ρu i u j [ ρũ j ( h + 12ũiũ i ) ) ũ i q j ] [ ρu j h 1 2 ρu j u i u i + ρu i τ ij ] + ũj 2 ρu i u i Turbulent Transport of Heat and Molecular Diffusion of Turbulent Energy: q tj = ρu j h, ρu i τ ij Turbulent Transport of Kinetic Energy: 1 2 ρu j u i u i ] 88

45 2.13 Turbulence Modelling Turbulence Modelling provides a mathematical framework for determining the additional terms (i.e., correlations) that appear in the FANS and RANS equations. Turbulence models may be classified as follows: Eddy-Viscosity Models (based on Boussinesq approxmiation) 0-Equation or Algebraic Models 1-Equation Models 2-Equation Models Second-Moment Closure Models Reynolds-Stress, 7-Equation Models 89