Governing Equations of Fluid Dynamics

Transcription

1 Chapter 3 Governing Equations of Fluid Dynamics The starting point of any numerical simulation are the governing equations of the physics of the problem to be solved. In this chapter, we first present the governing equations of fluid dynamics and their nondimensionalization. Then, we describe their transformation to generalized curvilinear coordinates. And finally, we close this chapter by presenting the governing equations for the case of an incompressible viscous flow. 3. Navier-Stokes System of Equations The equations governing the motion of a fluid can be derived from the statements of the conservation of mass, momentum, and energy [5]. In the most general form, the fluid motion is governed by the time-dependent three-dimensional compressible Navier-Stokes system of equations. For a viscous Newtonian, isotropic fluid in the absence of external forces, mass diffusion, finite-rate chemical reactions, and external heat addition, the strong conservation form of the Navier-Stokes system of equations in compact differential form can be written as ρ + (ρu) = t (ρu) + (ρuu) = p + τ t (ρe t ) + (ρe t u)=k T p u +( τ) u t This set of equations can be rewritten in vector form as follows Q t + E i x + F i y + G i z = E v x + F v y + G v z (3.) where Q is the vector of the conserved flow variables given by 34

2 3.. NAVIER-STOKES SYSTEM OF EQUATIONS ρ ρu Q = ρv ρw ρe t (3.2) and E i = E i (Q), F i = F i (Q) and G i = G i (Q) are the vectors containing the inviscid fluxes in the x, y and z directions and are given by ρu ρv ρw ρu 2 + p E i = ρuv ρuw, F ρvu i = ρv 2 + p ρvw, G ρwu i = ρwv ρw 2 + p (ρe t + p) u (ρe t + p) v (ρe t + p) w where u is the velocity vector containing the u, v and w velocity components in the x, y and z directions and p, ρ and e t are the pressure, density and total energy per unit mass respectively. The vectors E v = E v (Q), F v = F v (Q) and G v = G v (Q) contain the viscous fluxes in the x, y and z directions and are defined as follows τ xx E v = τ xy τ xz uτ xx + vτ xy + wτ xz q x τ yx F v = τ yy τ yz uτ yx + vτ yy + wτ yz q y τ zx G v = τ zy τ zz uτ zx + vτ zy + wτ zz q z where the heat fluxes q x,q y and q z are given by the Fourier s law of heat conduction as follows (3.3) (3.4) q x = k T x q y = k T y q z = k T z (3.5) and the viscous stresses τ xx, τ yy, τ zz, τ xy, τ yx, τ xz, τ zx, τ yz and τ zy, are given by the following 35

3 CHAPTER 3. GOVERNING EQUATIONS OF FLUID DYNAMICS relationships where µ is the laminar viscosity. τ xx = 2 ( 3 µ 2 u x v y w ) z τ yy = 2 ( 3 µ 2 v y u x w ) z τ zz = 2 ( 3 µ 2 w z u x v ) y ( u τ xy = µ y + v ) x ( u τ xz = µ z + w ) x ( v τ yz = µ z + w ) y τ yx = τ xy τ zx = τ xz τ zy = τ yz (3.6) Examining closely equations eq. 3., eq. 3.2, eq. 3.3 and eq. 3.4 and counting the number of equations and unknowns, we clearly see that we have five equations in terms of seven unknown flow field variables u, v, w, ρ, p, T, and e t. It is obvious that two additional equations are required to close the system. These two additional equations can be obtained by determining relationships that exist between the thermodynamic variables (p, ρ, T, e i ) through the assumption of thermodynamic equilibrium. Relations of this type are known as equations of state, and they provide a mathematical relationship between two or more state functions (thermodynamic variables). Choosing the specific internal energy e i and the density ρ as the two independent thermodynamic variables, then equations of state of the form are required. p = p (e i,ρ), T = T (e i,ρ) (3.7) For most problems in aerodynamics and gasdynamics, it is generally reasonable to assume that the gas behaves as a perfect gas (a perfect gas is defined as a gas whose intermolecular forces are negligible), i.e., p = ρr g T (3.8) where R g is the specific gas constant and is equal to 287 m2 for air. Assuming also that the s 2 K working gas behaves as a calorically perfect gas (a calorically perfect gas is defined as a perfect gas with constant specific heats), then the following relations hold e i = c v T, h = c p T, γ = c p c v, c v = R g γ, 36 c p = γr g γ (3.9)

4 3.2. NONDIMENSIONALIZATION OF THE GOVERNING EQUATIONS where γ is the ratio of specific heats and is equal to.4 for air, c v the specific heat at constant volume, c p the specific heat at constant pressure and h is the enthalpy. By using eq. 3.8 and eq. 3.9, we obtain the following relations for pressure p and temperature T in the form of eq. 3.7 p =(γ ) ρe i, T = p ρr g = (γ ) e i R g (3.) where the specific internal energy per unit mass e i = p/(γ )ρ is related to the total energy per unit mass e t by the following relationship, e t = e i + 2 ( u 2 + v 2 + w 2) (3.) In our discussion, it is also necessary to relate the transport properties (µ, k) to the thermodynamic variables. Then, the laminar viscosity µ is computed using Sutherland s formula µ = C T 3 2 (T + C 2 ) (3.2) where for the case of the air, the constants are C = ms and C K 2 =.4K. The thermal conductivity, k, of the fluid is determined from the Prandtl number (Pr =.72 for air) which in general is assumed to be constant and is equal to kg k = c pµ Pr (3.3) where c p and µ are given by equations eq. 3.9 and eq. 3.2 respectively. The first row in eq. 3. corresponds to the continuity equation. Likewise, the second, third and fourth rows are the momentum equations, while the fifth row is the energy equation in terms of total energy per unit mass. The Navier-Stokes system of equations eq. 3., eq. 3.2, eq. 3.3 and eq. 3.4, is a coupled system of nonlinear partial differential equations (PDE), and hence is very difficult to solve analytically. There is no general closed-form solution to this system of equations; hence we look for an approximate solution of this system of equation in a given domain D with prescribed boundary conditions D and given initial conditions DŮ. If in eq. 3. we set the viscous fluxes E v =, F v = and G v =, we get the Euler system of equations, which governs inviscid fluid flow. The Euler system of equations is a set of hyperbolic equations while the Navier-Stokes system of equations is a mixed set of hyperbolic (in the inviscid region) and parabolic (in the viscous region) equations. Therefore, time marching algorithms are used to advance the solution in time using discrete time steps. 3.2 Nondimensionalization of the Governing Equations The governing fluid dynamic equations shown previously may be nondimensionalized to achieve certain objectives. The advantage in doing this is that, firstly, it will provide conditions upon 37

5 CHAPTER 3. GOVERNING EQUATIONS OF FLUID DYNAMICS which dynamic and energetic similarity may be obtained for geometrically similar situations. Secondly, by nondimensionalizing the equations appropriately, the flow variables are normalized so that their values fall between certain prescribed limits such as zero and one. Thirdly, the procedure of nondimensionalization, also allows the solution to be independent of any system of units and helps to reduce the sensitivity of the numerical algorithm to round-off-errors. And finally, by nondimensionalizing the governing equations, characteristic parameters such as Mach number, Reynolds number and Prandtl number can be varied independently. Among many choices, in external flow aerodynamics it is reasonable to normalize with respect to the freestream parameters so that x = x L, ỹ = y L, z = z L ũ = u, ṽ = v, w = w U U U ρ = ρ T p, T =, p = ρ T t = tu L, ρ U 2 ẽ t = e t U 2, µ = µ µ (3.4) where denotes nondimensional quantities, the subscript denotes freestream conditions, L is some dimensional reference length (such as the chord of an airfoil or the length of a vehicle), and U is the magnitude of the freestream velocity. The reference length L is used in defining the nondimensional Reynold s number, this parameter represents the ratio of inertia forces to viscous forces, and is given by = ρ U L µ (3.5) where the freestream laminar viscosity µ is computed using the freestream temperature T according to eq When dealing with high speed compressible flow, it is also useful to introduce the Mach number. The Mach number is a nondimensional parameter that measures the speed of the gas motion in relation to the speed of sound a, a = Then the Mach number M is given by, [( ) ] p 2 = γ p ρ s ρ = γr g T (3.6) M = U a = U γ (p/ρ) = U γrg T (3.7) Finally, the remaining nondimensional quantities are defined as follows 38

6 3.2. NONDIMENSIONALIZATION OF THE GOVERNING EQUATIONS R g = R g U /T 2 = c p = (γ ) M 2 T C /2 = C µ C 2 = C 2 T γm 2 (3.8) Now, by simple replacing into the governing equations eq. 3. the dimensional quantities by their corresponding nondimensional equivalent, the following nondimensional equations are obtained Q t + Ẽi x + F i ỹ + G i z = Ẽv x + F v ỹ + G v z (3.9) where Q is the vector of the nondimensional conserved flow variables given by ρ ρũ Q = ρṽ ρ w ρẽ t (3.2) and Ẽi = Ẽi(Q), F i = F i (Q) and G i = G i (Q) are the vectors containing the nondimensional inviscid fluxes in the x, ỹ and z directions and are given by ρũ ρṽ ρ w ρũ 2 + p Ẽ i = ρũṽ ρũ w, ρṽũ Fi = ρṽ 2 + p ρṽ w, ρ wũ Gi = ρ wṽ ρ w 2 + p ( ρẽ t + p)ũ, ( ρẽ t + p)ṽ, ( ρẽ t + p) w (3.2) and Ẽv = Ẽv(Q), F v = F v (Q) and G v = G v (Q) are the vectors containing the nondimensional viscous fluxes in the x, ỹ and z directions and are given by 39

7 CHAPTER 3. GOVERNING EQUATIONS OF FLUID DYNAMICS τ xx Ẽ v = τ xy τ xz ũ τ xx +ṽ τ xy + w τ xz q x τ yx F v = τ yy τ yz ũ τ yx +ṽ τ yy + w τ yz q y τ zx G v = τ zy τ zz ũ τ zx +ṽ τ zy + w τ zz q z (3.22) However, in the process of nondimensionalizing the equations, the terms M and arises from the nondimensional viscous flux vectors. Therefore, the definition of the heat flux components and the viscous stresses may be modified as follows µ q x = (γ ) M Re 2 L Pr µ q y = (γ ) M Re 2 L Pr µ q z = (γ ) M Re 2 L Pr T x T ỹ T z (3.23) and 4

8 3.3. TRANSFORMATION OF THE GOVERNING EQUATIONS TO GENERALIZED CURVILINEAR COORDINATES τ xx = 2 ( µ 2 ũ 3 x ṽ ỹ w ) z τ yy = 2 ( µ 2 ṽ 3 ỹ ũ x w ) z τ zz = 2 ( µ 2 w 3 z ũ x ṽ ) ỹ τ xy = µ ( ũ ỹ + ṽ ) x τ xz = µ ( ũ z + w ) x τ yz = µ ( ṽ z + w ) ỹ (3.24) τ yx = τ xy τ zx = τ xz τ zy = τ yz Finally, by nondimensionalizing the equations of state eq. 3., we obtain p =(γ ) ρẽ i, T = p ρ R g = (γ ) ẽ i R g (3.25) where the nondimensional specific internal energy per unit mass ẽ i = p/(γ ) ρ is related to the nondimensional total energy per unit mass ẽ t by the following relationship, ẽ t =ẽ i + 2 (ũ2 +ṽ 2 + w 2) (3.26) Note that the nondimensional form of the equations given by eq. 3.9, eq. 3.2, eq. 3.2 and eq are identical (except for the ) to the dimensional form given by equations eq. 3., eq. 3.2, eq. 3.3 and eq For the sake of simplicity, the notation will be dropped for the remainder of this dissertation. Thus, all the equations will be given in nondimensional form unless otherwise specified. 3.3 Transformation of the Governing Equations to Generalized Curvilinear Coordinates The Navier-Stokes system of equation (eq. 3., eq. 3.2, eq. 3.3 and eq. 3.4) are valid for any coordinate system. We have previously expressed these equations in terms of a Cartesian coordinate system. For many applications it is more convenient to use a generalized curvilinear coordinate system. The use of generalized curvilinear coordinates implies that a distorted region in physical space is mapped into a rectangular region in the generalized curvilinear coordinate space (figure 3.). Often, the transformation is chosen so that the discretized equations are solved in a uniform logically rectangular domain for 2D applications and an equivalent uniform logically hexahedral domain for 3D applications. The transformation shall be such that there is a one-to-one correspondence of the grid points from the physical space (Cartesian coordinates) 4

9 CHAPTER 3. GOVERNING EQUATIONS OF FLUID DYNAMICS to computational space (generalized curvilinear coordinates). Figure 3.: Correspondence between the physical space (Cartesian coordinates) and the computational space (generalized curvilinear coordinates). Hereafter, we will describe the general transformation of the nondimensional Navier-Stokes system of equations (eq. 3.9, eq. 3.2, eq. 3.2 and eq. 3.22) given in the previous section between the physical space (Cartesian coordinates) and the computational space (generalized curvilinear coordinates). The governing equations are written in strong conservation form and expressed in terms of the generalized curvilinear coordinates as independent variables, thus the computations are performed in the generalized curvilinear coordinate space. The governing equations of fluid dynamics are transformed from the physical space P = P(x, y, z, t) to the computational space C = C(ξ, η, ζ, τ) by using the following transformations τ = τ (t) =t ξ = ξ (x, y, z, t) η = η (x, y, z, t) ζ = ζ (x, y, z, t) (3.27) where τ is considered to be equal to t and thus the transformation with respect to time is simple defined as τ = t as shown in eq Applying the chain rule, the partial derivatives of any quantity φ = φ(x, y, z, t) with respect to the Cartesian coordinates can be written as 42

10 3.3. TRANSFORMATION OF THE GOVERNING EQUATIONS TO GENERALIZED CURVILINEAR COORDINATES φ t = φ τ + ξ φ t ξ + η φ t η + ζ φ t ζ φ x = ξ φ x ξ + η φ x η + ζ φ x ζ φ y = ξ φ y ξ + η φ y η + ζ φ y ζ φ z = ξ φ z ξ + η φ z η + ζ φ z ζ (3.28) Then the governing equations may be transformed from physical space P to computational space C by replacing the Cartesian derivatives by the partial derivatives given in eq. 3.28, where the terms ξ x,η x,ζ x,ξ y,η y,ζ y,ξ z,η z,ζ z,ξ t,η t and ζ t are called metrics (they represents the ratio of arc lengths in the computational space C to that of the physical space P) and where ξ x represents the partial derivative of ξ with respect to x, i.e. ξ/ x, and so forth. Figure 3.2: Transformation from physical space to computational space. Left: structured grid in physical space. Right: logically uniform grid in computational space. In most cases, the transformation eq from physical space P to computational space C is not known analytically, rather it is generated numerically by a grid generation scheme. That is, we usually are provided with just the x, y and z coordinates of the grid points and we numerically generate the metrics using finite differences. The metrics ξ x,η x,ζ x,ξ y,η y,ζ y,ξ z,η z,ζ z,ξ t,η t and ζ t appearing in eq can be determined in the following manner. First, we write down the differential expressions of the inverse of the transformation eq. 3.27, dt = t τ dτ + t ξ dξ + t η dη + t ζ dζ dx = x τ dτ + x ξ dξ + x η dη + x ζ dζ dy = y τ dτ + y ξ dξ + y η dη + y ζ dζ dz = z τ dτ + z ξ dξ + z η dη + z ζ dζ (3.29) where the inverse of the transformation eq is 43

11 CHAPTER 3. GOVERNING EQUATIONS OF FLUID DYNAMICS t = t (τ) =τ x = x (ξ, η, ζ, τ) y = y (ξ, η, ζ, τ) z = z (ξ, η, ζ, τ) (3.3) and recalling that for a grid that is not changing (moving, adapting or deforming) t τ = and t ξ = t η = t ζ = dt = dτ thus Expressing eq in matrix form, we obtain dt dτ dx dy = x τ x ξ x η x ζ dξ y τ y ξ y η y ζ dη (3.3) dz z τ z ξ z η z ζ dζ In a like manner, we proceed with the transformation eq. 3.27, and we obtain the following differential expressions dτ = dt dξ = ξ t dt + ξ x dx + ξ y dy + ξ z dz dη = η t dt + η x dx + η y dy + η z dz dζ = ζ t dt + ζ x dx + ζ y dy + ζ z dz (3.32) which can be written in matrix form as dτ dt dξ dη = ξ t ξ x ξ y ξ z dx η t η x η y η z dy (3.33) dζ ζ t ζ x ζ y ζ z dz By relating the differential expressions eq of the transformation eq to the differential expressions eq. 3.3 of the transformation eq. 3.3, so that the metrics can be found, we conclude that ξ x,η x,ζ x,ξ y,η y,ζ y,ξ z,η z,ζ z,ξ t,η t,ζ t ξ t ξ x ξ y ξ z η t η x η y η z = x τ x ξ x η x ζ y τ y ξ y η y ζ ζ t ζ x ζ y ζ z z τ z ξ z η z ζ (3.34) 44

12 3.3. TRANSFORMATION OF THE GOVERNING EQUATIONS TO GENERALIZED CURVILINEAR COORDINATES This yields the following metrics relationships ξ x = J x (y η z ζ y ζ z η ) ξ y = J x (x ζ z η x η z ζ ) ξ z = J x (x η y ζ x ζ y η ) ξ t = (τ t x τ ξ x + τ t y τ ξ y + τ t z τ ξ z ) η x = J x (y ζ z ξ y ξ z ζ ) η y = J x (x ξ z ζ x ζ z ξ ) η z = J x (x ζ y ξ x ξ y ζ ) η t = (τ t x τ η x + τ t y τ η y + τ t z τ η z ) ζ x = J x (y ξ z η y η z ξ ) ζ y = J x (x η z ξ x ξ z η ) ζ z = J x (x ξ y η x η y ξ ) ζ t = (τ t x τ ζ x + τ t y τ ζ y + τ t z τ ζ z ) (3.35) For ξ t,η t and ζ t the following values are obtained after some manipulation ξ t = J x [x τ (y ζ z η y η z ζ )+y τ (x η z ζ x ζ z η )+z τ (x ζ y η x η y ζ )] η t = J x [x τ (y ξ z ζ y ζ z ξ )+y τ (x ζ z ξ x ξ z ζ )+z τ (x ξ y ζ x ζ y ξ )] ζ t = J x [x τ (y η z ξ y ξ z η )+y τ (x ξ z η x η z ξ )+z τ (x η y ξ x ξ y η )] (3.36) In eq and eq. 3.36, J x is the determinant of the Jacobian matrix of the transformation defined by or J x = (ξ, η, ζ) (x, y, z) J x = x ξ (y η z ζ y ζ z η ) x η (y ξ z ζ y ζ z ξ )+x ζ (y ξ z η y η z ξ ) (3.37) which can be interpreted as the ratio of the areas (volumes in 3D) in the computational space C to that of the physical space P. Once relations for the metrics and for the Jacobian of the transformation are determined, the governing equations eq. 3.9 are then written in strong conservation form as ˆQ t + Êi ξ + ˆF i η + Ĝi ζ = Êv ξ + ˆF v η + Ĝv ζ (3.38) where 45

13 CHAPTER 3. GOVERNING EQUATIONS OF FLUID DYNAMICS ˆQ = Q J x Ê i = J x (ξ t Q + ξ x E i + ξ y F i + ξ z G i ) ˆF i = J x (η t Q + η x E i + η y F i + η z G i ) Ĝ i = J x (ζ t Q + ζ x E i + ζ y F i + ζ z G i ) (3.39) Ê v = J x (ξ x E v + ξ y F v + ξ z G v ) Ê v = J x (η x E v + η y F v + η z G v ) Ê v = J x (ζ x E v + ζ y F v + ζ z G v ) The viscous stresses given by eq in the transformed computational space are ˆτ xx = 2 µ [2 (ξ x u ξ + η x u η + ζ x u ζ ) (ξ y v ξ + η y v η + ζ y v ζ ) (ξ z w ξ + η z w η + ζ z w ζ )] ˆτ yy = 2 µ [2 (ξ y v ξ + η y v η + ζ y v ζ ) (ξ x u ξ + η x u η + ζ x u ζ ) (ξ z w ξ + η z w η + ζ z w ζ )] ˆτ zz = 2 µ [2 (ξ z w ξ + η z w η + ζ z w ζ ) (ξ x u ξ + η x u η + ζ x u ζ ) (ξ y v ξ + η y v η + ζ y v ζ )] (3.4) ˆτ xy =ˆτ yx = µ (ξ y u ξ + η y u η + ζ y u ζ + ξ x v ξ + η x v η + ζ x v ζ ) ˆτ xz =ˆτ zx = µ (ξ z u ξ + η z u η + ζ z u ζ + ξ x w ξ + η x w η + ζ x w ζ ) ˆτ yz =ˆτ zy = µ (ξ z v ξ + η z v η + ζ z v ζ + ξ y w ξ + η y w η + ζ y w ζ ) and the heat flux components given by eq in the computational space are µ ˆq x = (γ ) M Re 2 L Pr (ξ xt ξ + η x T η + ζ x T ζ ) µ ˆq y = (γ ) M Re 2 L Pr (ξ yt ξ + η y T η + ζ y T ζ ) µ ˆq z = (γ ) M Re 2 L Pr (ξ zt ξ + η z T η + ζ z T ζ ) (3.4) Equations eq and eq are the generic form of the governing equations written in strong conservation form in the transformed computational space C (see [4], [85] and [8] for a detailed derivation). The coordinate transformation presented in this section, follows the same development proposed by Viviand [22] and Vinokur [2], where they show that the governing equations 46

14 3.4. SIMPLIFICATION OF THE NAVIER-STOKES SYSTEM OF EQUATIONS: INCOMPRESSIBLE VISCOUS FLOW CASE of fluid dynamics can be put back into strong conservation form after a coordinate transformation has been applied. Comparing the original governing equations eq. 3.9, eq. 3.2, eq. 3.2 and eq and the transformed equations eq and eq. 3.39, it is obvious that the transformed equations are more complicated than the original equations. Thus, a trade-off is introduced whereby advantages gained by using the generalized curvilinear coordinates are somehow counterbalanced by the resultant complexity of the equations. However, the advantages (such as the capability of using standard finite differences schemes and solving the equations in a uniform rectangular logically grid) by far outweigh the complexity of the transformed governing equations. One final word of caution. The strong conservation form of the governing equations in the transformed computational space C is a convenient form for applying finite difference schemes. However, when using this form of the equations, extreme care must be exercised if the grid is changing (that is moving, adapting or deforming). In this case, a constraint on the way the metrics are differenced, called the geometric conservation law or GCL (see [5], [55] and [85]), must be satisfied in order to prevent additional errors from being introduced into the solution. 3.4 Simplification of the Navier-Stokes System of Equations: Incompressible Viscous Flow Case Equations eq. 3., eq. 3.2, eq. 3.3 and eq. 3.4 with an appropriate equation of state and boundary and initial conditions, governs the unsteady three-dimensional motion of a viscous Newtonian, compressible fluid. In many applications the fluid density may be assumed to be constant. This is true not only for liquids, whose compressibility may be neglected, but also for gases if the Mach number is below.3 [6, 53]; such flows are said to be incompressible. If the flow is also isothermal, the viscosity is also constant. In this case, the dimensional governing equations in primitive variable formulation (u, v, w, p) and written in compact conservative differential form reduce to the following set u t (u) = + (uu) = p ρ + ν 2 u where ν is the kinematic viscosity and is equal ν = µ/ρ. The same set of equations in nondimensional form is written as follows u t (u) = + (uu) = p + 2 u which can be also written in nonconservative form (or advective/convective form [6]) u t u = + u u = p + 2 u or in expanded three-dimensional Cartesian coordinates 47

15 CHAPTER 3. GOVERNING EQUATIONS OF FLUID DYNAMICS u x + v y + w z = u t + u u x + v u y + w u z = p x + ( 2 ) u x u y u z 2 v t + u v x + v v y + w v z = p x + ( 2 ) v x v y v z 2 w t + u w x + v w y + w w z = p x + ( 2 ) w x w y w z 2 (3.42) This form (the advective/convective form), provides the simplest form for discretization and is widely used when implementing numerical methods for solving the incompressible Navier-Stokes equations, as noted by Gresho [6]. Equation eq governs the unsteady three-dimensional motion of a viscous, incompressible and isothermal flow. This simplification is generally not of a great value, as the equations are hardly any simpler to solve. However, the computing effort may be much smaller than for the full equations (due to the reduction of the unknowns and the fact that the energy equation is decoupled from the system of equation), which is a justification for such a simplification. The set of equations eq can be rewritten in vector form as follow Q t + E i x + F i y + G i z = E v x + F v y + G v z (3.43) where Q is the vector containing the primitive variables and is given by Q = u v (3.44) w and E i, F i and G i are the vectors containing the inviscid fluxes in the x, y and z directions and are given by u E i = u 2 + p uv, F i = uw v vu v 2 + p vw, G i = w wu wv w 2 + p (3.45) The viscous fluxes in the x, y and z directions, E v, F v and G v respectively, are defined as follows E v = τ xx τ xy, F v = τ yx τ yy, G v = τ zx τ zy (3.46) τ xz τ yz τ zz Since we made the assumptions of an incompressible flow, appropriate nondimensional terms and 48

16 3.4. SIMPLIFICATION OF THE NAVIER-STOKES SYSTEM OF EQUATIONS: INCOMPRESSIBLE VISCOUS FLOW CASE expressions for shear stresses must be used, these expressions are given as follows τ xx = 2 u x τ yy = 2 v y τ zz = 2 w z τ xy = ( u y + v ) x τ xz = ( w x + u ) z τ yz = ( w y + v ) z (3.47) τ yx = τ xy τ zx = τ xz τ zy = τ yz Following the procedure presented in the previous section, the nondimensional incompressible Navier-Stokes system of equations eq in the computational space C is expressed as ˆQ t + Êi ξ + ˆF i η + Ĝi ζ = Êv ξ + ˆF v η + Ĝv ζ (3.48) where ˆQ = Q J x Ê i = J x (ξ x E i + ξ y F i + ξ z G i ) ˆF i = J x (η x E i + η y F i + η z G i ) Ĝ i = J x (ζ x E i + ζ y F i + ζ z G i ) (3.49) Ê v = J x (ξ x E v + ξ y F v + ξ z G v ) ˆF v = J x (η x E v + η y F v + η z G v ) Ĝ v = J x (ζ x E v + ζ y F v + ζ z G v ) In eq. 3.49, ˆQ is the vector containing the primitive variables and Êi, ˆF i and Ĝi are the vectors containing the inviscid fluxes in the ξ, η and ζ directions respectively, and are given by 49

17 CHAPTER 3. GOVERNING EQUATIONS OF FLUID DYNAMICS U ˆQ = u J x v, Ê i = uu + pξ x J x vu + pξ y, w wu + pξ z V W ˆF i = uv + pη x J x vv + pη y, Ĝ i = uw + pζ x J x vw + pζ y wv + pη z ww + pζ z (3.5) where U, V and W are the contravariant velocities U = uξ x + vξ y + wξ z, V = uη x + vη y + wη z, W = uζ x + vζ y + wζ z The shear stresses given by eq expressed in the computational space C are as follow τ xx = 2 (ξ x u ξ + η x u η + ζ x u ζ ) τ yy = 2 (ξ y v ξ + η y v η + ζ y v ζ ) τ zz = 2 (ξ z w ξ + η z w η + ζ z w ζ ) τ xy = (ξ y u ξ + η y u η + ζ y u ζ + ξ x v ξ + η x v η + ζ x v ζ ) τ xz = (ξ z u ξ + η z u η + ζ z u ζ + ξ x w ξ + η x w η + ζ x w ζ ) τ yz = (ξ y w ξ + η y w η + ζ y w ζ + ξ z v ξ + η z v η + ζ z v ζ ) τ yx = τ xy (3.5) τ zx = τ xz τ zy = τ yz Substituting the expressions for the shear stresses given by eq. 3.5 into the viscous flux vectors Ê v, ˆF v and Ĝv (given by eq. 3.49) in the ξ, η and ζ directions respectively, we obtain the following equations 5

18 3.4. SIMPLIFICATION OF THE NAVIER-STOKES SYSTEM OF EQUATIONS: INCOMPRESSIBLE VISCOUS FLOW CASE Ê v = ˆF v = Ĝ v = a u ξ + b u η c v η + c 2 w η + b 2 u ζ d v ζ + d 2 w ζ J x a v ξ + c u η + b v η c 3 w η + d u ζ + b 2 v ζ d 3 w ζ a w ξ c 2 u η + c 3 v η + b w η d 2 u ζ + d 3 v ζ + b 2 w ζ a 2 u η + b u ξ + c v ξ c 2 w ξ + b 2 u ζ e v ζ + e 2 w ζ J x a 2 v η c u ξ + b v ξ + c 3 w ξ + e u ζ + b 3 v ζ e 3 w ζ a 2 w η + c 2 u ξ c 3 v ξ + b w ξ e 2 u ζ + e 3 v ζ + b 3 w ζ a 3 u ζ + b 2 u ξ + d v ξ d 2 w ξ + b 3 u η + e v η e 2 w η J x a 3 v ζ c 4 u ξ + b 2 v ξ + d 3 w ξ e u η + b 3 v η + e 3 w η a 3 w ζ + d 2 u ξ d 3 v ξ + b 2 w ξ + c 8 u η e 3 v η + b 3 w η (3.52) where a = ξx 2 + ξy 2 + ξz, 2 a 2 = ηx 2 + ηy 2 + ηz, 2 a 3 = ζx 2 + ζy 2 + ζz 2, b = ξ x η x + ξ y η y + ξ z η z, b 2 = ξ x ζ x + ξ y ζ y + ξ z ζ z, b 3 = ζ x η x + ζ y η y + ζ z η z, c = ξ x η y η x ξ y, c 2 = η x ξ z ξ x η z, c 3 = ξ y η z η y ξ z, d = ξ x ζ y ζ x ξ y, d 2 = ζ x ξ z ξ x ζ z, d 3 = ξ y ζ z ζ y ξ z, (3.53) e = η x ζ y ζ x η y, e 2 = ζ x η z η x ζ z, e 3 = η y ζ z ζ y η z equations eq and eq written in a more compact way, can be expressed as Ê v = ( ξ ξ) u ξ +( ξ η) u η +( ξ ζ) u ζ J x ( ξ ξ) v ξ +( ξ η) v η +( ξ ζ) v ζ ( ξ ξ) w ξ +( ξ η) w η +( ξ ζ) w ζ ˆF v = ( η ξ) u ξ +( η η) u η +( η ζ) u ζ J x ( η ξ) v ξ +( η η) v η +( η ζ) v ζ ( η ξ) w ξ +( η η) w η +( η ζ) w ζ Ĝ v = ( ζ ξ) u ξ +( ζ η) u η +( ζ ζ) u ζ J x ( ζ ξ) v ξ +( ζ η) v η +( ζ ζ) v ζ ( ζ ξ) w ξ +( ζ η) w η +( ζ ζ) w ζ (3.54) Equation eq. 3.48, together with eq. 3.49, eq. 3.5 and eq. 3.54, are the governing equations of an incompressible viscous flow written in strong conservation form in the transformed computational space C. Hence, we look for an approximate solution of this set of equations in a given domain D with prescribed boundary conditions D and given initial conditions DŮ. So far, we have just presented the governing equations; in the following chapters the grid generation method as well as the numerical scheme for solving the governing equations will be explained. 5