Lecture 5.7 Compressible Euler Equations

Lecture 5.7 Compressible Euler Equations

Nomenclature Density u, v, w Velocity components p E t H u, v, w e S=c v ln p - c M Pressure Total energy/unit volume Total enthalpy Conserved variables Internal energy/unit mass Ratio of specific heats Entropy Speed of sound Mach number u/c

Vector Form of Euler Equations U E F G t x y z 0 (5.7.) where u w u u p u uw w uw w w p E t Et p u Et p Et p w U E u F p G w Equation of state has to be added to close the system 3

Equation of State Assuming the fluid as a perfect gas one can write p= RT where R is the gas constant, we have RT Et = + u + v + w g - ( ) g is the ratio of the specific heats (.4 for air) 4

Euler s Equations in D To illustrate the flux vector splitting, consider the one dimensional equivalent of Eq. (5.7.), that is, U E + = 0 t x æ ö u æu ö æ r ö æ ru ö u U = u ru, E ru p p = = + = + u çu 3 re t ruet pu è ø çè ø èç + ø uu 3 u p ç + çè u u ø (5.7.) 5

Elimination of the Pressure p For perfect gas, we have R p= rrt = ( g- ) re (after using cv = & e= ct v ) g - E= e+ u re= re- ru æ ö æ u ö p= ( g-) re ru ( g ) u - = - - èç ø è ø 3 ç u æ ö u æ u = r ö u æ u ö E= ( g ) u + - 3 - u u ç, u ru çè = ø ç u3 re è = t ø uu æ 3 u ö u + ( g -) u 3 - u ç u è ç çè øu ø (5.7.3) 6

Modified Equations Rewrite Eq. (5.7.) as U U + A = 0 t x æ ö 0 0 E u where A= = ( g-3) (3-g) u g- U 3 3 ç ( g-) u -guet - ( g- ) u + ge gu çè ø (5.7.4) 7

Conservative Form of A The conservative form A is æ ö 0 0 u u A = ( g 3) (3 g) g - - - u u u uu 3 u gu gu ( g-) -g - ( g- ) + ç çè u u u u u ø 3 3 3 3 (5.7.5) 8

Equations in Primitive Variables We have from Eq. (5.7.4) U U V U V E U + A = 0 + = 0 t x U t U U x V V E U V V V + = 0 + B = 0 t U U V x t x æ æu r 0ö ö r æ ö V E U B 0 u, for V u = = ç =ç U U V r p èç ø ç 0 rc u è ç è ø ø (5.7.6) Here, c = p/ is the speed of sound 9

Equations in Characteristic Variables The D Euler s equations can be also written as æu 0 0ö W W æ c c ö + C = 0, C= 0 u c 0, W S u u - = - + t x g g ç è - - ø ç è0 u+ c 0 ø T (5.7.7) where S is the entropy. The elements of the W are called Riemann-invariants as they invariant along the characteristics defined by respectively. dx/dt=u, dx/dt=u-c and dx/dt=u+c (5.7.8) 0

Hyperbolic Nature Since the matrices A, B & C are similar, the eigenvalues of these matrices are u-c, u, u+c, which are distinct for c 0. Therefore, the corresponding eigenvectors are linearly independent. Hence Euler s equations are hyperbolic in nature. Further, there exists a non-singular matrix P such that P AP = C (5.7.9)

Boundary Conditions Assume that the initial conditions are given and the boundary conditions, required for the Euler s equations, are to be identified in the following way: For hyperbolic systems, the number of boundary conditions required at any point of the boundary must be equal to the number of incoming characteristics at that point. Therefore, for supersonic inflow (u>0, u c>0), three boundary conditions on U are required at the inflow.

For subsonic inflow (u>0, u-c<0), only two conditions must be given. For subsonic outflow (u<0, u+c>0), only one condition and finally for supersonic outflow (u<0, u+c<0), no conditions must be given. A general practice is to prescribe u and e at the inflow boundary and the pressure p at a supersonic inflow and subsonic outflow boundaries. 3

Nature of Solutions The solutions of Euler s equations can have Shocks Contact discontinuities Simple waves Expansion fans 4

Shocks A shock will form whenever the characteristics intersect as shown in Fig. 5.7. Note: Characteristics originating from dx/dt=u-c and dx/dt=u+c only can intersect because the lines originating from dx/dt=u are the path lines. Fig. 5.7. Intersection of characteristics 5

Rankine-Hugoniot Conditions Across a shock we must have ( ) su - u = u -u r l r l u u su u p p r l r- l = - + r- l u r u l ( ) ( ) su - u = u H-u H 3r 3l r r l l (5.7.0) where the subscripts l and r represent, the left and right sides of the shock and u =,u = u, u 3 = E t are the conserved variables. 6

Entropy Condition g - R S pr - Sl = ln -g p ( ) r l ln r r r l (5.7.) Where, S is the entropy defined by S = c v ln p - which states that, the entropy does not decrease as it traverse through a shock. This condition is also equivalent to say that the fluid particle can enter a shock from the supersonic side. Please note that a shock always separates a supersonic and subsonic states. 7

Contact Discontinuities Since the lines originating from dx/dt=u can t intersect, they don t produce shocks. Since S is constant along these lines but differ between lines, they can support a jump in S. The discontinuity has to travel with the speed s = u l =u r. That is, the fluid particles on either side of this discontinuity have the same speed, therefore, the particles which are initially in contact will remain in contact. 8

Discretization Finite difference/volume discretizations of Eq. (5.7.) is similar to the corresponding discretizations discussed in incompressible flow computations. However, due to hyperbolic nature of these equations, certain algebraic manipulations can be attempted in the discretization of the flux computations. One of them is the flux vector splitting, but before discretising flux vector splitting let us look at few more simple numerical schemes. 9

Lax-Friedrich s Scheme For D Euler s equations, Lax-Friedrich s first order scheme is given by U E + = 0 t x t U = U + U - E -E i=,,3, N -, n= 0,,, x ( - + ) ( + -) n+ n n n n i i i i i x (5.7.) where, x = t/ x, t is the step length in time direction x isthe step length in space direction, N x is the number of cells. The stability requirement is t ( + ) (5.7.) x u c 0

Lax-Friedrich s Scheme For D Euler s equations Lax-Friedrich s first order scheme is given by U E F + + = 0 t x y n+ n n n n Ui, j = ( Ui, j-+ Ui-, j+ Ui+, j+ Ui, j+ ) 4 t t x n n y n n - ( Ei+, j-ei-, j) - ( Fi, j+ -Fi, j- ) i=,,3, N -, j=,,3, N -, n= 0,,, x y (5.7.3) where, x = t/ x, y = t/ y, N x and N y are the number of cells in x and y directions, respectively. The stability requirement is tx( u+ c) + ty( v+ c) (5.7.4)

Lax-Wendroff Scheme Lax-Wendroff second order scheme for D Euler s equations is given U E F + + = 0 t x y t t U = U + U + U + U - E -E - F -F 4 U = U -t E -E - F -F x y ( - - + + ) ( + - ) ( + - ) n+ / n n n n n n n n i, j i, j i, j i, j i, j i, j i, j i, j i, j ( + - ) t ( + - ) n+ n n+ / n+ / n+ / n+ / i, j i, j x i, j i, j y i, j i, j i=,,3, N -, j=,,3, N -, n= 0,,, x y (5.7.5) The stability condition is given by (when x = y) Dt Dx ( v c) + (5.7.6)

Summary of Lecture 5.7 Mathematical and computational aspects to solve compressible Euler equations are discussed in this lecture. END OF LECTURE 5.7 3