Mathematical modeling is now a key tool of research into dynamic processes in continuum mechanics. Each particular problem is solved with already existing or newly developed models and methods, whose properties are determined from a priori study into stability, approximation, monotonicity etc within linear approaches. The accuracy of di erence schemes is mainly evaluated through comparison between calculated results and reference solutions. The paper discusses some problems which have analytical solutions. These are shock convergence, the dynamic compression of a gas sphere, and some problems with stationary shocks.
The properties of di erence schemes that approximate the conservation laws are often evaluated with a priori methods which involve studies into stability, approximation, monotonicity, conservatism, distraction etc. It should however be noted that most of these methods are developed for acoustic approximations and simple equations of state. In continuum mechanics, the properties of a mathematical model may notably di er from what linear theory predicts due to nonlinearities induced by real-world equations of state, shocks, plasticity and other material properties.
Linear theory looses its rigor when applied to nonlinear equations. The importance of the convergence theorem [1] is strongly exaggerated because it is still proved for linear equations and not for nonlinear ones, and real calculations are done not for vanishing but finite x and t. A very vivid discussion of stability and convergence criteria and their rigidity can be found in [2].
The calculation of shock waves strong discontinuities in all material properties requires special attention. On the shock surface the conservation laws take the form of nonlinear algebraic equations which relate the values of quantities across the shock. Entropy jumps as all the other functions do. This is the fundamental di erence between a shock and a wave where the quantities vary continuously. Flows with shocks are often simulated with homogeneous methods which treat the strong shock as a layer of a finite width comparable with the cell size. This ability of di erence schemes is called distraction [3]. Since the states across the shock are related by the Hugoniot, there must be a mechanism which allows entropy to grow in the shock distraction region. Physical viscosity and heat conduction in continuum mechanics equations cannot give a distraction width of several cell sizes. The proposal by Neumann and Richtmyer to use a mathematical ’viscosity’ [4] seems to resolve the problem. Their method has gained wide acceptance. Pseudo-viscosity is taken in di erent forms linear, quadratic, or linearquadratic [4-7]. But the method does not ensure convergence to the exact solution if the form of pseudo-viscosity changes. So, the author of [8] gives an example where di erent schemes with di erent viscosities converge to di erent solutions in the limit. In [9], there is an example of spherical convergence where energy dissipation defined by pseudo-viscosity is shown to be several times higher than energy dissipation due to plasticity.
Advantages and disadvantages of a mathematical model can be seen from comparison between its predictions and analytical solutions. The paper discusses some problems which have exact analytical solutions. It gives their statements (initial and boundary conditions, equations of state, and physical parameters) and solutions in the form of formulas or tables. In order to verify performance of a di erence scheme, one needs to solve the problem numerically and compare the result with the exact solution.
The problems are broken into two groups for stationary and non-stationary shocks. In problems with stationary shocks, derivatives in the exact solution are zero everywhere beyond the distraction zone and hence approximation errors are also zero. In the distraction zone, the derivatives and approximation errors reach high values. Here the strong shock is smeared over several cells where entropy di ers. These problems help verify real shock distraction, monotonicity, entropy variation, and the dependence of calculated results on the relation between steps in space and time, and on cell size (the number of points in the mesh). All these properties reveal themselves di erently for strong and weak shocks. Shock strength is characterized be the di erence between pressures behind and before the shock.
In problems with non-stationary shocks, the derivatives and derivative-dependent approximation errors are high beyond the distraction zone. This group includes shock convergence and spherical shell convergence problems. In the last problem, the boundary conditions and released energy are adjusted so as to keep material density constant despite large pressure and velocity gradients.
Some analytical solutions are used for comparison with results obtained with the di erence schemes which are based on the energy dissipation method described in [1013]. 2
Consider a material with parameters P0, V0, E0, U0 which do not change with time. At
a time t0 its left boundary instantaneously starts moving at a constant positive velocity,
producing a shock wave which propagates into the material. The equations
gas in the form
quantities
relate the material states P− = P0; V− = V0; E
− = E0; U
− = U0 with the state after
the discontinuity P = P+; V = V+; E = E+; U = U+ before and behind the shock, and
the shock velocity W. The number of quantities is larger than the number of equations
(
PV = ( − 1) E
and transform equations (
U2 V0 + √︃ ︃( + 1 U2 )︃2 4
V0 +
P0 V0
U2;
where U = U − U0. From equations (
WU+ − P+ = WU− − P− ;
W"+ − P+U+ = W"− − P− U−
W = (P − P0) / (U − U0); V = V0 − (U − U0) /W; E = E0 +0; 5 (P + P0) (V − V0) : (
P = (n − 1) E + C02k ( − 0k) ; where
= 1=V - is material density, 0k - is its density at a point with coordinates
T = 0, P = 0 and C0k - is sound velocity at this point. For EOS (
P = P0 +
0 U2 + n + 1 4 √︃ ︃( + 1 4 schemes x0 is a Lagrangian coordinate. At t > 0, U = 1 is specified on the left boundary (x0 = 0) and U = 0 is on the right one (x0 = 1).
The quantities behind the shock front and front velocity W are determined from
equations (
= 4, E = 0:5, P = 4=3, W = 4=3. At t = 0:375, the shock is at a
point x0 = 0:5 and the analytical solution is determined by
(
P = 0, = 1:0, E = 0, U = 0, for x0 > 0:5 Figures 1 and 2 depict P(x0) and U(x0) at t = 0:575. The solid lines show analytical solutions and the marked ones show calculations with the di erence scheme from [12]. The calculations were done with a uniform mesh of N = 100 points in x0 and Courant number 0.5.
P 1.2 0.8 0.4 0 0.4 0.45 0.5 0.55 x0 0.4 0.45 0.5 0.55
Problem 2. Strong shock in monatomic gas described by EOS (
The quantities behind the shock front and front velocity are determined from equations
(
P = 1:125, = 9:0, E = 0:5, U = 1 for x0 ≤ 0:5, and
P = 0, = 1:0, E = 0, U = 0, for x0 > 0:5.
Figures 3 and 4 depict P(x0) and U(x0) at t = 0:44444. The solid lines show analytical solutions and the marked ones show calculations with the di erence scheme from [12]. The calculations were done with a uniform mesh of N = 100 points in x0 and Courant number 0.5.
Problem 3. The weak shock wave in monatomic gas. At t = 0, a region 0 ≤ x0 ≤ 1
is occupied with monatomic ideal gas described by EOS (
U
0.2
0
0.3
0.3
0.4
0.5
0.4
0.5
Problem 4. Strong shock in condense matter. At t = 0, a region 0 ≤ x0 ≤ 1 is
occupied with condense matter described by EOS (
P = (n − 1) E + − 1;
(
P = 0.8 0.4 @E @V
+ P = 0: (
Equations (
There no incompressible matter in nature. Mechanics simply considers a wide class of flows where density remains constant in time. Density constancy is often understood as incompressibility, i.e., s = 0 and C2 = ∞. It is not true. The property of flow is not the property of matter.
0.45 0.5 0.55 0.45 0.5 0.55
P = 4:0, = 1:0, E = 2:0, U = 0 for x0 > 0:5. Figures 9 and 10 depict P(x0) and U(x0) at t = 0:105448. The solid lines show analytical solutions and the marked ones show calculations with the di erence scheme from [13]. The calculations were done with a uniform mesh of N = 100 points in x0 and Courant number 0.5. 3
The problem of bubble collapse in fluid, or spherical shell convergence, arises in
connection with cavitation corrosion of propellers. Solutions to the problem can be found
in [14-16]. The full continuum mechanics model for 1D spherically symmetric flow
of ideal compressible continua includes mass conservation, motion and internal energy
equations:
As a rule [16], the models of ’incompressible’ fluid do not include the energy
conservation law and the equation of state. This makes them internally contradictive. As
follows from (
E;
Equations (
Here we limit ourselves to flows where specific volume is independent of either M, or t, i.e., V = const. Also, we assume that fluid is compressible, i.e., its compressibility S is nonzero.
For V = V0, the equation that relates the Eulearian coordinate r and the Lagrangian coordinate dM = (︁ 4 r2⧸︁V0)︁ dr can be integrated from M = 0 at r = rB to an arbitrary finite M where UB is bubble boundary velocity.
Find UB from (17) and substitute in the bubble boundary motion equation r2BUB = f (t) ;
U = UB rr2B2 :
︃( drB )︃
dt M
= UB:
Integrate (18) together with (15), to obtain the dependence of rB on t
Here rB is the time dependent coordinate of the bubble boundary. At V = V0, equation
(
r2U = f (t) :
Since f is independent of M, equations (15) and (16) are valid for arbitrary M. On the
bubble boundary where M = 0, equation (16) takes the form
(
: t0 It is seen from (17) and (19) that the motion of the bubble boundary is completely defined by f (t). If f (t) < 0, then UB < 0 too, i.e., the bubble collapses. The boundary convergence time tf is found from (19) at rB = 0 The values Ua0 and Pa0 are found from (22) and (23) at t = t0. In Lagrangian coordinates the pressure, velocity and released energy are defined by ⎛︃( ra0 )︃3 ⎜ Ua = UB0 ⎜⎜⎜⎝⎜ rB0 t − t0 ⎞⎟⎟− 2/3
⎟ − t f − t0 ⎟⎠⎟
; Pa =
UB20 ⎛⎜⎜⎜︃( t f − t )︃− 4/3
⎜ 2V0 ⎜⎝⎜ t f − t0
− ⎜⎜⎜⎜⎜⎝⎛︃( rrBa00 )︃3 − (︃ ttf − − tt00 )︃⎟⎟⎟⎟⎠⎟⎞− 4/3⎟⎟⎟⎟⎟⎠⎟⎞ : P =
UB20 ⎛⎜⎜(︃ t f − t )︃− 4/3
⎜ 2V0 ⎜⎝⎜ t f − t0 − ⎜⎜⎜⎝⎜⎛ ttff − − tt0 + 3V0 M ⎞− 4/3⎞ 4 r3B0 ⎟⎟⎠⎟⎟ ⎟⎠⎟⎟⎟⎟ ; U = UB0 ⎜⎜⎜⎝⎛⎜ ttff − − tt0 + 3V0 M ⎞− 2/3 ⎟ ⎟ 4 r3B0 ⎟⎠⎟
; Let all quantities on the outer boundary be subscripted ”a”. Assume that the shell mass is equal to Ma. The coordinate of the outer boundary, ra, relate to that of the inner boundary rB as ra = ︁( r3B + b)︁ 1/3 ; where b = 34V0 Ma = ra30 − r3B0. Pressure and velocity on the outer boundary are (19) (20) (21) (22) (23) (24) As the reference problem we consider the motion of 10% shell.
Problem 6. The motion of a 10%-shell. At t0 = 0, rB0 = 1, ra0 = 1:1, V0 = 1, and Ma = 1:38649. The velocity of the inner boundary is UB0 = 1. The EOS of shell material with parameters 0k = 1, C0k = 1 and = 2 has the form P = 2 E + − 1. At the initial time, pressure, specific internal energy and velocity in the shell are defined by For t ≥ 0 and Ma ≥
0 the solution has the form dq dt M ≥
︃( = 1 − 3t + 3M )︃− 7/3
− (1 − 3t)− 7/3 : P (M) = 1 − ︃( 0, energy release as a function of t Fig. 11. Problem 6. 4
In di erent years, there were published a number of papers [18-22] with self-similar solutions to shock convergence in infinite ideal gas. Shock convergence in a gas sphere of finite radius is considered in [23,24]. At t = t0, pressure in gas is P0 = 0, density 0 = const, velocity U0 = 0, and specific internal energy E0 = 0. The boundary of the sphere is at a point (r0; t0). Velocity on the boundary is Ug0 < 0. In other words, velocity jumps on the boundary, producing a shock wave which moves into the sphere. At the time when the shock converges, t f , its coordinate rw is zero. The equation of motion which satisfies all these conditions is (25) (26) (27) (28) (29) (30) (31) for n > 0 Its di erentiation gives shock velocity where
rw = r0 t f − t0
︃( t f − t )︃n D = D0 t f − t0 ︃( t f − t )︃n− 1
; D0 = − r0n⧸︁ (︁ t f − t0 :
︁)
For solving the problem we change from the variables t and r to variables t and (t; r). The function (t; r) is taken such as to remain constant on the shock. Its simplest form reads as = r (︃ t f − t0 )︃n r 0 t f − t : Now express P, and U as functions of time multiplied by functions of : P = p (t) ( ) ; = (t) ( ) ;
U = u (t) M ( ) :
( ) ; ( ) ; M ( ) such that to allow them at = 1 take the values + 1 With these w, w, Mw the function
, u and p take the forms 2 (n − 1) n + ′ (M − n ; For M′, ′, ′ , these equations give a system of linear homogeneous equations. If its determinant
Z = (M − ) ︁( (M − ) 2)︁ of coe is nonzero, the system has a unique solution. At the point * where Z = 0, the matrix cients and the augmented matrix of coe cients should be considered. At this point their ranks are identical and equal to 2, and all third-order minors are zero, hence the system of equations (33) and (34) has a unique solution. It is easy to show that with the zero third-order minors we come to (n − 1) * (2 (M* − * ) −
M* ) + 2 nM* (M* − * ) = 0: The value of n is found from the condition that equation (36) holds simultaneously with After appropriate manipulation for conversion to the functions , , M and variables t, , we obtain equations for functions which only depend on : (32) (33) (34) (35) (36) (37)
From equations (36) and (37) we find the appropriate values of n for each . This solution was used to evaluate the accuracy of some shock calculation methods.
Problem 7. A cold gas sphere of radius rg0 = 1 with parameters P0 = 0, 0 = 1, U0 = 0, Ug0 = 1, from t, and
= 5=3. The boundary condition is defined through reverse transition , , M to t, M and P, , U. Pressure and boundary velocity as functions of time are presented in Table 1. Pressure, density and velocity profiles at t = 0:4; 0:45; 0:5 (marked 1, 2, 3) are shown in Figs. 13-15. The solid lines show the analytical solution derived in this work, the lines with circles show calculations by the VOLNA code [25] with no shock smearing, and the dashed lines show VOLNA calculations with shock smearing. The calculations were done on a uniform mesh of 100 points in r at t = t0. In Fig.14, entropy traces are seen in the dashed density profiles, which are a result of shock smearing on the boundary. Figure 16 depicts M( ), ( ) and ( ) for 1 ≤
Problem 7. The boundary condition in the case of U -1 -2 -3 δ 0.2 0.4
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