AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 31 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Linearization and Characteristic Relations 1 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 31 Outline 1 Non Conservation Form and Jacobians 2 Linearization Around a Localized Flow Condition 3 Hyperbolic Requirement 4 Characteristic Relations 5 Application to the One-Dimensional Euler Equations 6 Boundary Conditions 2 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 3 / 31 Non Conservation Form and Jacobians Consider again an equation written in conservation law form where t + F (W ) = S F (W ) = ( F T x (W ) F T y (W ) F T z (W ) ) T In three dimensions, the above equation can be re-written as follows or where t + F x(w ) x + F y (W ) t y + F z(w ) z = S + A x + B y + C z = S (1) A = F x(w ), B = F y (W ), C = F z(w ) are called the Jacobians of F x, F y, and F z with respect to W, respectively 3 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 4 / 31 Non Conservation Form and Jacobians For example for the Euler equations in two dimensions, each of the Jacobians is a 4 4 matrix In general for m-dimensional vectors W = (W 1 W m ) T and F = (F 1 F m ) T F = F 1 F 1 m 1..... F m 1 F m m 4 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 5 / 31 Non Conservation Form and Jacobians If W = W (V ), Eq. (1) can be transformed as follows V t + A V x + B V y + C V z = S (2) where A = T 1 AT, B = T 1 BT, C = T 1 CT, S = T 1 S and T = V represents the Jacobian of W with respect to V The Jacobians with respect to W are then given by = V V = V T 1 5 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 6 / 31 Non Conservation Form and Jacobians Definition: G(W 1,, W m ) is said to be a homogeneous function of degree p, where p is an integer, if s > 0 G(sW 1,, sw m ) = s p G(W 1,, W m ) Example: A linear function is a homogeneous function of degree 1 s > 0, G(sW 1,, sw m ) = sg(w 1,, W m ) Exercise: show that for a perfect gas, the fluxes F x, F y, and F z of the Euler equations are homogeneous functions (of W ) of degree 1 A homogeneous function of degree p has scale invariance that is, it has some properties that remain constant when looking at them either at different length- or time-scales 6 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 7 / 31 Non Conservation Form and Jacobians Theorem 1 (Euler s theorem): A differentiable function G(W 1,, W m ) is a homogeneous function of degree p if and only if m G W i (W 1,, W m ) = pg(w 1,, W m ) i i=1 Proof: ( ) differentiate definition with respect to s and set s = 1 ( ) define g(s) = G(sW 1,, sw m ) s p G(W 1,, W m ), differentiate g(s) to get an ordinary differential equation in g(s), note that g(1) = 0, and conclude that g(s) = cst = 0 7 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 8 / 31 Non Conservation Form and Jacobians Theorem 2: If G(W 1,, W m ) is differentiable and homogeneous of degree p, then each of its partial derivatives G i (for i = 1,, m) is a homogeneous function of degree p 1 s 0, G p 1 G (sw 1,, sw m ) = s (W 1,, W m ) i i Proof: Straightforward (differentiate both sides of the definition with respect to W i ) 8 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 9 / 31 Linearization Around a Localized Flow Condition Linearization can be either physically relevant (small perturbations), convenient for analysis, or useful for constructing a linear model problem in either case, it leads to a linear problem For the purpose of constructing a linear model version of Eq. (1), the coefficient matrices A, B, and C of this equation are often simply frozen to their values at a local flow condition designated by the subscript o and represented by the fluid state vector W o, which leads to t + A o x + B o y + C o z = S o (3) The above linear equation can be insightful for the construction or analysis of a CFD scheme 9 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 10 / 31 Linearization Around a Localized Flow Condition On the other hand, the genuine linearization of Eq. (1) (with S independent of W ) about a flow equilibrium condition W o which is physically more relevant leads to the following equation t + A o x + B o y + C o z S ow + A ow o x + B ow o y + C ow o z = 0 (4) Hence, the following remarks are noteworthy: in a genuine linearization such as in Eq. (4), W is a perturbation around W o which should be denoted in principle by δw in a genuine linearization around a dynamic equilibrium condition, the source term does not contribute a frozen right hand-side in general, Eq. (4) and Eq. (3) are different however, if the linearization is performed about a uniform flow condition W o and S = 0, Eq. (4) and Eq. (3) become identical 10 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 11 / 31 Linearization Around a Localized Flow Condition Consider here the linear model equation (3) t + A o x + B o y + C o z = S o Linearized equations such as the above equation have exact solutions Let W (x, y, z, t 0 ) denote an initial value for W at time t 0 : this initial condition can be expanded by Fourier decomposition with wave numbers k xj, k yj, and k zj as follows W (x, y, z, t 0 ) = I (x, y, z) = j c j e i(kx j x+ky j y+kz j z) In this case, the exact solution of Eq. (3) for t > t 0 is W (x, y, z, t) = j e i(t t0 )(kx j Ao +ky j Bo +kz j Co ) cj e i(kx j x+ky j y+kz j z) + (t t 0 )S o 11 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 12 / 31 Linearization Around a Localized Flow Condition W (x, y, z, t) = j e i(t t0 )(k xj A o+k yj B o+k zj C o) c j e i(kx j x+ky j y+kz j z) +(t t 0 )S o Hence, the solution of Eq. (3) has both a linear growth term and, depending on the eigenvalues of the matrix M j = k xj A o + k yj B o + k zj C o a possible exponential growth in time components 12 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 13 / 31 Hyperbolic Requirement Consider the following equation G x α + H x 1 = 0 (5) For example for the unsteady Euler equations in one dimension x α = t, x 1 = x, G = W = (ρ ρv x E) T H = F x = ( ρv x ρv 2 x + p (E + p)v x ) T For the steady Euler equations in two dimensions x α = x, x 1 = y G = F x ( ρvx ρv 2 x + p ρv x v y (E + p)v x ) T H = F y = ( ρv y ρv x v y ρv 2 y + p (E + p)v y ) T 13 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 14 / 31 Hyperbolic Requirement Let A = H G and let Λ = diag(λ 1,, λ m ) be the diagonal matrix of eigenvalues λ 1,, λ m of A Eq. (5) is hyperbolic if (1) λ k is real for each k (2) A has a complete set of eigenvectors A is diagonalizable that is Q / A = H G = Q 1 ΛQ In the general multidimensional case ( see Eq. (1 ) ), the system is hyperbolic if the matrix M = k x A + k y B + k z C has only real eigenvalues and a complete set of eigenvectors, for all sets of real numbers (k x, k y, k z ) 14 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 15 / 31 Characteristic Relations In mathematics, the method of characteristics is a technique for solving partial differential equations Essentially, it reduces a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface It is applicable to any hyperbolic partial differential equation, but has been developed mostly for first-order partial differential equations Characteristic theory is pertinent to the treatment of boundary conditions and CFD schemes such as flux split schemes (Steger and Warming) and flux difference vector splitting schemes (Roe) 15 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 16 / 31 Characteristic Relations Consider the following unsteady homogeneous hyperbolic equations written in non conservation form t + A x A is diagonalizable and therefore = 0, A = F = A(W ) (6) A = Q 1 ΛQ (7) where Λ = diag (λ 1,, λ m ) = Λ(W ) and Q = Q(W ) Let r i denote the i-th column of Q 1 : AQ 1 = Q 1 Λ Ar i = λ i r i r i is A s i-th right eigenvector Let l i denote the i-th column of Q T which is the i-th row of Q: QA = ΛQ A T Q T = Q T Λ A T l i = λ i l i (or li T A = λ i li T ) l i is A s i-th left eigenvector 16 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 17 / 31 Characteristic Relations Substituting (7) into Eq. (6) and pre-multiplying by Q leads to the so-called characteristic form of Eq. (6) Q t + ΛQ x = 0 The characteristic variables ξ = (ξ 1 ξ m ) T are defined as follows (note the differential form) dξ = QdW Substituting the definition of the characteristic variables in the characteristic form of the governing equations leads to ξ t + Λ ξ x = 0 (8) which is also called the characteristic form of the governing equations and has the advantage of decoupling the characteristic variables 17 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 18 / 31 Characteristic Relations Each characteristic equation within Eq. (8) can be written as ξ i (1 λ i ) T = 0, i = 1,, m which shows that ξ i t + λ ξ i i is a directional derivative in the x t x plane in the direction (1 λ i ) T Then, in English, each characteristic equation says that there is no change in the solution ξ i in the direction of (1 λ i ) T in the x t plane Now, consider a curve x = x(t) that is everywhere tangent to (1 λ i ) T in the x t plane: Then, the slope of the vector (1 λ i ) T is the slope of the curve x = x(t) and is given by dx dt = λ i 18 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 19 / 31 Characteristic Relations Then Eq. (8) is equivalent to the following: dξ i = 0 ( or ξ i = cte) for dx dt = λ i, i = 1,, m This is a wave solution: the eigenvalues λ i are wave speeds, and the wavefronts dx dt = λ i are sometimes also called characteristic curves (or simply characteristics) 19 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 20 / 31 Application to the One-Dimensional Euler Equations t + F x x = 0, W = (ρ ρv x E) T, F x = ( ρv x ρvx 2 ) T + p (E + p)v x ( E ρ v x 2 ), and the speed of sound c given by 2 with p = (γ 1) c 2 = γp ρ Choose V = (ρ v x p) T as the fluid state vector (with primitive variables) and re-write the governing equations in non conservation form ( see Eq. (1) and Eq. (2) ) V t V + A x = 0, A = v x ρ 0 0 v x 1 ρ 0 ρc 2 v x 20 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 21 / 31 Application to the One-Dimensional Euler Equations Diagonalize the resulting hyperbolic equations A = Q 1 ΛQ QA Q 1 = Λ Λ = Q = v x 0 0 0 v x + c 0 0 0 v x c 1 0 1 0 1 c 2 1 ρc 0 1 1 ρc Q 1 = 1 0 0 ρ 2c 1 2 ρc 2 ρ 2c 1 2 ρc 2 21 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 22 / 31 Application to the One-Dimensional Euler Equations Let ξ = (ξ 0 ξ + ξ ) T denote the characteristic variables The three characteristic equations are ξ + t ξ t ξ 0 t + v ξ 0 x x + (v x + c) ξ + x + (v x c) ξ x = 0 = 0 = 0 with in this case dξ = QdV Therefore, these equations are equivalent to dξ 0 = dρ dp c 2 = ds = 0 for dx = v xdt (s denotes here the entropy) dξ + = dv x + dp ρc = 0 for dx = (v x + c)dt dξ = dv x dp ρc = 0 for dx = (v x c)dt 22 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 23 / 31 Application to the One-Dimensional Euler Equations Hence, the solution of these characteristic equations can be written as ξ 0 = s = cst for dx = v x dt (entropy wave) dp ξ + = v x + ρc = cst for dx = (v x + c)dt (acoustic wave) dp ξ = v x ρc = cst for dx = (v x c)dt (acoustic wave) Notice that in this case, only the first characteristic equation is fully analytically integrable (but not its corresponding dx = v x dt) For this and other reasons, characteristics are important conceptually, but not of too great importance quantitatively (9) 23 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 24 / 31 Application to the One-Dimensional Euler Equations 24 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 25 / 31 Application to the One-Dimensional Euler Equations Integral curves of the characteristic family note that dξ = QdV dv = Q 1 dξ: the i-th column of Q 1 (i = 1, 2, 3), denoted here by r i, is the i-th right eigenvector of the Jacobian matrix A and defines a vector field it follows that dv = Q 1 dξ = r i dξ i (10) i hence, one can look for a set of states V (η) that connect to some starting state V 0 through integration along one of these vector fields these constitute integral curves of the characteristic family two states V a and V b belong to the same j-characteristic integral curve (dv = r j dξ j = dr j ) if they are connected via the integral V b = V a + b a dr j (11) 25 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 26 / 31 Application to the One-Dimensional Euler Equations Integral curves of the characteristic family (continue) consider now the case of a linear hyperbolic equation with a constant advection matrix the state vector V can be decomposed in eigen components a j-characteristic integral curve in state-space is a set of states for which only the component along the r j eigenvector varies, while the components along the other eigenvectors may be non zero but should be non varying for a nonlinear hyperbolic equation, the decomposition of V is no longer a useful concept, but the integral curves are the nonlinear equivalent of this idea 26 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 27 / 31 Application to the One-Dimensional Euler Equations Riemann invariants one can express integral curves not only as integrals along the eigenvectors of the Jacobian ( as in Eq. (11) ), but also curves on which some special scalars are constant ( as in Eq. (10) with only one dξ j 0 and two dξ i = 0 see Eqs. (9) ) in the 3D parameter space of V = (V 1, V 2, V 3) = (ρ, v x, p), each curve is defined by two of such scalars such scalar fields are called Riemann invariants of the characteristic family here, ξ + and ξ are the Riemann invariants of the 1-characteristic integral curve ξ 0 and ξ are the Riemann invariants of the 2-characteristic integral curve ξ 0 and ξ + are the Riemann invariants of the 3-characteristic integral curve note: the 2- and 3-characteristic integral curves represent here acoustic waves which, if they do not topple to become shocks, preserve entropy: Hence, entropy (ξ 0 ) is a Riemann invariant of these two families 27 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 28 / 31 Application to the One-Dimensional Euler Equations Riemann invariants (continue) hence, one can regard these integral curves now as the crossing lines between the two contour curves of the two Riemann invariants the value of each of the two Riemann invariants now identifies each of the characteristic integral curves 28 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 29 / 31 Application to the One-Dimensional Euler Equations Riemann invariants (continue) in summary, the Riemann invariants are: constant along characteristic integral curves of the partial differential equation mathematical transformations made on a system of first-order partial differential equations to make them more easily solvable 29 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 30 / 31 Application to the One-Dimensional Euler Equations Simple waves note that if the Riemann invariants are constant along the characteristic curve dx dt = λ i, all flow properties are constant along this characteristic curve definition: a wave is called a simple wave if all states along the wave lie on the same integral curve of one of the characteristic families hence, one can say that a simple wave is a pure wave in only one of the eigenvectors examples a simple wave in the 1-characteristic family (dv = r 1 dξ 0 ) is a wave (or region of the flow) in which v x = cst and p = cst but the entropy s can vary a simple wave in the 3-characteristic family (dv = r 3 dξ ) is for example an infinitesimally weak acoustic wave in one direction in Chapter 5, situations will be encountered where a contact discontinuity and a rarefaction wave are simple waves 30 / 31
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 31 / 31 Boundary Conditions 31 / 31