tait pure water
TRANSCRIPT
Pergamon Mathl. Comput. Modelling Vol. 25, No. 3, pp. 25-36, 1997
Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved
0895-7177197 $17.00 + 0.00 PII: s0895-7177(97)00013-7
A New Averaging Scheme for the Riemann Problem in Pure Water
TZE- JANG CHEN Department of Applied Mathematics
Feng-Chia University, Taiwan, R.O.C.
C. H. COOKE Department of Applied Mathematics and Statistics
Old Dominion University, Norfolk, VA 23529, U.S.A.
(Received September 1996; accepted November 1996)
Abstract-The numerical investigation of shock phenomena in gas or liquid media where enthalpy is the preferred thermodynamic variable poses special problems. When an expression for internal energy is available, the usual procedure is to employ a splitting scheme to remove source terms from the Euler equations, then upwind-biased shock capturing algorithms are built around the Riemann problem for the conservative system which remains. However, when the governing equations are formulated in terms of total enthalpy, treatment of a pressure time derivative as a source term leads to a Riemann problem for a system where one equation is not a conservation law. The present, research establishes that successful upwind-biased shock capturing schemes can be based upon the pseud+conservative system. A new averaging scheme for solving the associated Riemann problem is developed. The method is applied to numerical simulations of shock wave propagation in pure water.
Keywords-Riemann problem in water, Roe averaging, Numerical solution of P.D.E., Numerical model.
1. INTRODUCTION
Previous numerical studies of underwater shock phenomena, for the most part, have employed the simplified Tait equation of state [I]. With this equation of state, Ho11 [2] has established that, to good approximation, a functional form equivalence exists between the shock relations for the ideal gas and water. Thus, the Riemann problems for the two media can be solved using identical methods (31. In this research, the problem of numerically calculating shock wave propagation in liquid media with Modified-Tait equation of state is considered. This gives rise to the difficulty that techniques for exact solution of the Riemann problem may be unknown.
A further complicating factor is the assumption (true particularly for water) that a relation specifying the dependence of internal energy upon the variables of state is unknown. However, when a relation between enthalpy and the variables of state is available, an enthalpy formulation for the Euler equations of compressible flow can be useful. Unfortunately, there appears in the energy equation a term involving the time derivative of pressure, which most conveniently can be treated as a source term. When source terms are removed by splitting, the resulting equations do not all represent conservation laws, although a divergence form is exhibited.
Due to the absence of energy conservation, it is clear that the pseudo-system does not fully describe shock dynamics. Still, it can be used as a base upon which to build effective shock capturing algorithms. Demonstration of this fact is the major contribution of the present research. It is shown that the piecewise linearization technique of Roe [4] can be employed. However, the
Typeset by A@-‘QX
25
26 T.-J. CHEN AND C. H. COOKE
averaging scheme which aids piecewise linearization must be carefully formulated. Although our aims and methods of accomplishing this by means of a new averaging scheme are similar to those of Glaister [5-81, the results are applicable under entirely different and unexpected circumstances.
For quality assurance, a well-known problem for the ideal gas is investigated. Next, the shock capturing method is applied to the Riemann problem for pure water, where the modified-Tait equation of state for sea water is used to approximate the state of pure water. Numerical solutions of the benchmark problem appear excellent.
2. CONSTITUENT EQUATIONS FOR PURE WATER (MODIFIED-TAIT EQUATION OF STATE)
The modified-Tait equation of state has been used [2] for predicting the hydrodynamic proper- ties of seawater at the front of a propagating shockwave. Ho11 [2] shows that when this equation is applicable, by making substitutions P* = P + B and y* = N, the Riemann problem for water can be approximately solved using well-known gas-dynamic methods. This equation has the form
N P=B p -1. 10 1 PO
Here B = B(T) is essentially constant, and p0 is the density, where P = P, with P, = 0. For pure water, the values B = 2959 bars (2922 atm) and N = 7.415 give the correct speed of sound [9] (4865 ft/sec) at 20’ C and 1 atm. The speed of sound is given by
NP $=---. P
3. GOVERNING EQUATIONS AND WEAK SOLUTION FOR PSEUDO-CONSERVATIVE SYSTEMS
The one-dimensional equations of motion for an inviscid, non-heat-conducting flow of a gas or liquid, where specific enthalpy, h, rather than specific internal energy, e, is the preferred thermodynamic variable, can be written in the form
ut “I- F(U), + W(U) = 0. (3)
Here,
with
and
P u= pu [ 1 PH PfJ F = p+pz?
[ 1 PHU
(4)
(5)
(6)
The unknowns are P, pressure; p, density; U, velocity; h, specific enthalpy; and H, specific total enthalpy (H = h + u2/2). The independent variables are t, time, and 2, streamwise distance.
Glaister [5] has analyzed the averaging method as a technique for piecewise linearization, for a fully conservative system where both the perfect gas law and a general equation of state are considered. Perhaps surprisingly, he arrives at identical expressions for the averages 0, although expressions for eigenvalues and eigenvectors are radically different. Hence, the averaging appears to be uninfluenced by the particular expression used for pressure. Thus, in view of the extremely complicated general equation of state for water, the approach to be used here is the following: an averaging scheme for the pseudo-conservative system (3)-(5) will be proposed in the sequel where the Tait equation of state will be employed. The full system (3)-(6) is to be solved by means of operator splitting.
3.1. An Approximate
Consider the Riemann
A New Averaging Scheme 27
Riemann Solver (Weak Formulation)
problem for the Pseudo-Conservative system of hyperbolic equations
U, + F, = 0. (7)
The nonlinear Riemann problem is to be approximately solved by means of the locally linearized Riemann problems
U~+A(UL,UR)+U~=O, 03)
U(x,O) = UL, if 2 < 0,
UR, if z > 0. (9)
Here A(UL, UR)+ satisfies the following.
1. A(UL, UR)Q is a constant matrix. 2. A(UL, UR)Q depends on the path Q(s; UL, UR). 3. GJ is the path which links the states UL, UR. 4. When s = 0, @(O; U,, UR) = UL. 5. When s = 1, (a(l; UL, UR) = UR. 6. A(UL, UR)+ satisfies
s
1
A(@(s; V,,UR))~(S;U,,UR) ds =A(UL,UR)+(UR- UL), (10) 0
AW U), = A(U),
where
A(u) = g.
7. A(UL, UR)+ has real eigenvalues and a complete set of eigenvectors.
For example, if the path is considered as the straight line which links UL
and
~(s;uL,uR)=uL+s(uR-UL), s E [O, 11,
1
AWL, uR)@ = s
A (UL + s (UR - UL)) ds. 0
Several extensions of Roe’s linearization to an arbitrary equation of state Grossman and Walters [lo] follow the original method that introduced a
have been proposed. parameter vector w
such that u and f(u) are both quadratic functions of w, to obtain a Roe-averaged matrix. Glaister [5-8,111 uses direct approximations of the eigenvalues and eigenvectors of A(uR, us) to provide linearizations valid for general convex equation of state. The approach of Liou [12] assumes that A(ILR, ILL) is the exact Jacobian matrix evaluated at some average state ti,
(11)
(12)
and UR,
(13)
(14)
A(UL,UR) =A(U). 05)
Since A(UL, UR) is hard to be directly calculated from (lo)-( 12), one can choose another vari- able fc(w) such that the Riemann solver in [13] is constructed by letting fc satisfy
1. fc is a smooth function, 2. fo(w~) = UL, fO(wR> = UR, and
3. Ao(w) = $$f is a regular matrix for 20.
Km 25-3-8
28 T.-J. CHEN AND C. H. COOKE
The path @e linking UL and UR satisfies
@‘o(s;uL, UR) = fO(WL +s(WR - WL)). (16)
From (16),
J 1
0 A(Qo (3; UL,UR))z (s;UL,UR) ds,
=(WR-WL) [lA(f~(W~+S(W~-W~)))A~(W~+s(W~-WL))ds. (17) JO
Since
J 1
UR-UL=(WR-WL) Ao(WL+~(WR-WL)) ds, 0
it is seen that WR-wL= UR-UL
J; Ao(WL + s(WR - WL)) ds'
where
Upon substituting (19) into (17) and comparing with (lo), it is easy to see that
A.+0 =CB-'.
Hence, from (17)-(19)
where
A(UL,UR)~,, = C(UL,UR)Q~B(UL,UR)&
B (UL,UR)+, = J 1
Ao(WL+~(WR-W~))ds, 0
and 1
c (UL,UR)a, = J A(~o(WL + s(WR - WL)))AO(WL +s(WR -WL)) ds. 0
(1%
(20)
(21)
(22)
(23)
Hence, one can compare Aap, = CB-l with A(U) in order to find the proper average for the primitive variables.
An outline of the method follows.
1. Given w= (Wr,?.L$ )...( WJT.
2. Resolve the state variable U into variable w such that
3. Find
4. Find
A =dfo 0 dW’
1
B (UL,UR)@,, = J Ao(WL +s(WR -WL))ds. 0
5. Find A(U)= g,
as a function A(u(w)).
6. Find
A New Averaging Scheme
A(u(w))Ao(w).
29
7. Find
s
1
c (UL, uR)+., = A (fo (WL + s (WR - WL))) Ao (WL + s (WR - WI,)) ds. 0
8. Find A@, = CB-l.
9. Compare Aa., = CB-’ with A(U) to find the proper average.
3.2. Pseudo-Conservation Form of the Euler Equations
The one-dimensional equations of motion for an inviscid, non-heat-conducting, flow of a gas or liquid, where specific enthalpy, h, rather than specific internal energy, e, is the preferred thermodynamic variable, can be written in the form
u, + F(U), + W(U) = 0. (24)
Here
with
and
F=
w=
PU P + pu2
PHU
0 0
pt 1
(25)
(26)
(27)
The unknowns are P, pressure; p, density; u, velocity; h, specific enthalpy; and H, specific total enthalpy (H = h + u2/2). The independent variables are t, time, and 2, streamwise distance.
The quantities p, u represent the one dimensional density and velocity which are functions of streamwise distance z and time t. The following identities hold:
PE=PH-P,
H=h+$
E=e+%,
N
p=p y -1 ) D 1 PO
(28)
where /3, ~0, and N are constants. In the state variables
u = (~1, ~2, wlT = (P, P, pWT 1 (29)
and
f(u) = (f1,.f2,h)~ = (P~,P+PU~+PH)~ = ( ) [(-f&)N-l]+$,y)T. (30) ‘~12 P
30 T.-J. CHEN AND C. H. COOKE
Also, U, + A(U)Uz = 0,
SX where A(U) is the Jacobian matrix aU,
0 1 0
iv N-l PNP; UI
4 _~ ‘$2 0 ‘111 ‘111 1.
(31)
(32) -u2u3 ‘113 u2
u1 T iy u1
Derivation of the averaging method can be outlined further as follows. 1. Given w = (Wl,W2,2~3)~ = Jis(l,~,H)~.
2. Resolve the state variable U into variable w such that to(w) = U(w). Hence,
3. Find Ao(w) = 2.
fO(W) = u(W).= (W:,WIWZIW~W~)~. (33)
8fO Ao(w) = - dW (34)
2Wl 0 0
= [ W‘J Wl 0 1 . (35) W3 0 Wl
4. Find B(UL,UR)+~ = s,’ Ao(WL +s(wR - WL))~.
where i = 1,2,3. Hence, 1
B= Ao (w~+s(wR--WL)) ds
[
2til 0 0 - -
= w2 Wl 0 a3 0 ?iIl 1 >
where
where i = 1,2,3.
B-’ =
loo SiFl
@2 1 -- - 2?i$ ,281
0
a3
_iq 0 -!-
a1
ds
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
A New Averaging Scheme 31
5. Find A(U) = $$, as a function of w such that A(u(w)).
I
0 1 o- 4
44~)) = oNpiN - 2 22 0
-WY% W3 W2 - -
WI Wl Wl-
6. Find A(u(w))Ao(w).
(46)
W2 Wl 0
A(‘L~(w))Ao(w) = 2pNp,N~;N-’ 2~2 o . 0 w3 W2 1
7. Find C(UL, UF& = Ji A(~o(WL + S(WR - WL)))AO(WL + S(WR - WL)) ds.
a2 ZiQ 0
2@Np,NGi;N-’ 2ti2 0 , 0 ti3 ti2 1
where
1 -2N-1
WI = SC ‘WlL + s (f&R - WlL) 2N-1 ds
0 >
wg - w 1L”
2N (WlR - WlL) ’ if WlR # WlL,
ZZ
2N- 1 WlL ’ if W1R = W1L.
8. Find Aa,, = CB-‘.
[
0 1 0 Aao = ~N,zI,~~~-’ --ii* 2ii 0 . -Hii IT ii 1
9. Compare AaPo = CB-’ with A(U) to find the proper average.
also,
p=
= N--l d PZ - Pf N k'R - PL) ’
hence for WR # WL, 2N
-2N-1 = ‘w 1
WiN - WL
2N (‘WR - WL) ’
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
32 T.-J. CHEN AND C. H. COOKE
4. EIGENVALUES AND EIGENVECTORS FOR THE PSEUDO-CONSERVATION EQUATIONS
The eigenvalues (Xl, X2, and Xs) and eigenvectors (ei, es, and es) for the medium pure water will now be indicated.
General Equation of State
Here,
au3 1
e2,3 =
[ 1 U+;g*fi .
Ii
T&t Equation of State
For the Tait equation of state the eigenvalues and eigenvectors are
Xl = u,
x2 = 21 + a,
and
X3 = u - a,
and
e2,3 =
[ 1 x2,3 . H
(56)
(57)
(58)
(59)
(61) (62)
(63)
(64)
(65)
5. THE NUMERICAL METHOD
The form (3)-(6) for the Euler equations is rarely employed when studying gas dynamic or other flow problems where a specifying relation for internal energy is known. However, in cir- cumstances where the constituent equations involve enthalpy rather than internal energy, this form is necessary. In the present research, the problem of adapting shock capturing techniques which have been useful in gas dynamics to the numerical calculation of shocked flows involving pure water, with general equation of state, is attempted. Constituent relations h = h(T) and
A New Averaging Scheme 33
P = P(p,T), though complicated, exist; however, expressions for internal energy are unknown to the authors.
Here, the usual approach is followed: the source term in equation (1) is removed through Sod’s operator splitting technique [14], and effective shock capturing methods are determined for the equation
u, + F(U), = 0, (66)
by means of the averaging approach previously outlined. In the second stage of the splitting, pressure effects are incorporated by solving
ut + W(U) = 0. (67)
Let A(U) be the Jacobian matrix of the flux function, and let xk; lc = 1, M be a uniform partition of a computational domain for solutions of (3)-(6), with Ax = x~+~ -xcj, and Aj+ilalJ = U 3+1 - Uj. Piecewise linearization as a technique for solving (3)-(5) leads to a sequence of linear Riemann problems
U, + A (0) U, = 0, (68)
u CxTtn) = [Ujlx < Xj+1/2; Uj+l,X > Xj+l/2] , (69)
which are to be solved using Roe’s scheme and the new averaging previously discussed. The numerical solution of equations (3)-(5) now proceeds via the upwind-biased shock captur-
ing scheme
uj n+l = Ujn + 2 (F*j+1/2 - F*j-112) . (70)
Here, one expression for the flux F* is
Fj+1/2 = F (Uj) + F (U,,,) - x$k lAk\ Rk
2 3 (71)
where the eigenvalues X and eigenvectors R are evaluated on the average value fi,,1,2. The constants LY result from the usual eigenvector projection.
6. COMPUTATIONAL EXPERIMENTS
Several numeral experiments have been accomplished, whose purpose is to assess effectiveness of the shock-capturing scheme previously indicated.
CASE I: THE IDEAL GAS. The Riemann problem for the ideal gas, whose solution is well known, is now considered. Results for the case in which pi = PL = 1.0, UL = 0.0, and PR = PR == 0.1, UR = 0.0 are reported in [15,16].
The first question to be settled concerns the necessity of iterating the second stage of the splitting, or incorporation of the source term. Figures 1 and 2 show significant differencfes in comparison of results for the pressure and velocity profiles, with and without iteration. As far as accuracy in general, the results obtained by iteration agree well with those of [15,16], in terms of shock location and strength. Three point shock resolution is about what is to be expected from first-order accuracy. Thus, there has been obtained a viable algorithm for numerically solving the Euler equations with total enthalpy the preferred thermodynamic variable.
The next issue concerns relative effectiveness of the averaging schemes proposed in Section 3.
CASE II: PURE WATER. For pure water, the Riemann problem to be considered is characterized by PL = 997.286, PR = 0.955 bar, with pr, = 1037.8, PR = 997.94 kg/m3, zero initial velocity, and temperature 25” C, 30” C. For the Tait equation of state, pc = 997.04796, N = 7.2, B = 2996.
34 T.-J. CHEN AND C. H. COOKE
1.5
-“.50 0.2 0.4 0.6 0.8 1
Strearuwise Distance Streanjwise Distance
1.5
1 x .Y 1.
I
~.._......_. ii
---..._.. 5.
Q 0.5 ‘.... “..... -.............-..-..
‘*.....__ I ‘.....
Oo 0.2 0.4 0.6 0.8 1
Streamwise Distance Streamwise Distance. Figure 1. The well-known Riemann problem for the ideal gas (noniterated), pr, = PL = 1.0, uL = 0.0, and PR = PR = 0.1, UR = 0.0, which are reported in [15,16].
1 4 _.._ . -..-._ . .._..... _..._.. ,:. . . . . . . . . . __.*._
3 .:.
2 3 _ Y..
$ 0.5 - - E 2- :..
E! X.
3 '.....
Da '.. l-
--------..-_-.
‘0 ii; 0.4 0.6 0.8 . 1 '0 0.2 0.4 0.6 0.8 '-i
Streamwise Distance Streamwise Distance
1 . ‘C.. 1~ . . :, ':* . . . . x . . . . . .
-5 0.5 - 's. '%., - zi 2 0.5 - :.. . . . . B
k.-......_._..,~ '..... . . N...-_..~~-.~~~ ~.m.."....-._ Ei
'0 0.2 0.4 0.6 0.8
Streamwise Distance 1 '0 0.2 0.4 0.6 0.8
Streamwise Distance 1
Figure 2. The well-known Riemann problem for the ideal gas (iterated), pr, = PL = 1.0, UL = 0.0, and PR = PR = 0.1, UR = 0.0, which are reported in [15,16].
Figures 3 and 4 show comparison of results for the solution at time 0.0002 seconds, for Tait equation of state. In Figure 3 and 4, density variations across the contact surface are too slight to be visible, although contact surface position is apparent.
Using the correspondence p = P + B, y = N, the Riemann problem (for the Tait equation of state) can be solved using a gas dynamics Remann solver. The results indicate that the contact surface speed (the maximum velocity), together with shock strengths, are reasonably what is to be expected.
7. CONCLUSIONS
The method of Roe and a new averaging scheme has been adapted to the problem of obtaining a first order accurate, upwind-biased shock capturing scheme for the medium pure water and Tait equation of state. With minor modification, the method could be applied to seawater problems.
A New Averaging Scheme 35
~________ ..I......_. __..._._..._..___
Streamwise Distance
I g800
I 0.2 0.4 0.6 0.8 1
Streamwise Distance
I OO
I 0.2 0.4 0.6 0.8 1
Streamwise Distance
:------.
8.4980 . . . . .._ __....___.:
0.2 0.4 0.6 0.8 1
Streamwise Distance
Figure 3. For pure water, the Riemann problem to be considered is characterized by Pr, = 997.286, PR = 0.955 bar, with pi = 1037.8, pi = 997.94 kg/m3, zero initial velocity, and temperature 25’ C. For the Tait equation of state, pe = 997.04796, N = 7.2, B = 2996.
Streamwise Distance
1040 )_-.__.. s
$ :I 1 ‘....____..___“.._.._.._ _ ..,,, ____._ .j
g800m
Streamwise Distance
0 0.2 0.4 0.6 0.8
Streamwise Distance
9 XW _ _..___. --..____ ._.,
1
8% 1 0.2 0.4 “0.6 0.8 1
Streamwise Distance Figure 4. For pure water, the Riemann problem to be considered is characterized by PL = 997.286, PR = 0.955 bar, with pi = 1037.8, PR = 997.94kg/m3, zero initial velocity, and temperature 25’ C. For the Tait equation of state, po = 997.04796, N = 7.2, B = 2996.
An averaging scheme which aids shock recognition and which allows Roe’s piecewise lineariza- tion method to be applied to the enthalpy oriented Euler equations (3)-(6) has been derived, and tested. It is seen that the new averaging can be applied, perhaps unexpectedly, to pseudo conservative, divergence form systems. Equivalent numerical results are obtained, for the new averaging and Roe-Glaister averaging. Availability of these averaging schemes now allows appli- cation of the excellent second-order accurate TVD scheme of Harten, by means of Strang’s [14] operator splitting, to calculation of shockwaves in water.
36 T.-J. CHEN AND C. H. COOKE
The method has been demonstrated, through application to typical Riemann problems for both the ideal gas and the medium pure water.
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