Formulations (Under Development)¶
The governing equations of the hydro-acoustic wave include the continuity equation
(1)
and the momentum equations
(2)
where is the density, and the Cartesian component of the velocity, the pressure, the Kronecker delta, the second viscosity coefficien, the dynamic viscosity coefficient, the time, and , and the Cartesian coordinate axes. Newtonian fluid is assumed.
The above four equations in (1) and (2) have five
independent variables , and , and hence are
not closed without a constitutive relation. In the bulk
package,
the constitutive relation (or the equation of state) of choice is
where is a constant and the bulk modulus. We chose to use the density as the independent variable, and integrate the equation of state to be
(3)
where and are constants. Substituting (3) into (2) gives
(4)
Jacobian Matrices¶
We proceed to analyze the advective part of the governing equations (1) and (4). Define the conservation variables
(5)
and flux functions
(6)
Aided by the definition of conservation variables in (5), the flux functions defined in (6) can be rewritten with , and
(7)
By using (5) and (7), the left-hand-side of the governing equations can be cast into the conservative form
(8)
Aided by using the chain rule, (8) can be rewritten in the quasi-linear form
(9)
where the Jacobian matrices , and are defined as
(10)
Aided by using (7), the Jacobian matrices defined in (10) can be written out as
(11)
Hyperbolicity¶
Hyperbolicity is a prerequisite for us to apply the space-time CESE method to a system of first-order PDEs. For the governing equations, (1) and (4), to be hyperbolic, the linear combination of the three Jacobian matrices of their quasi-linear for must be diagonalizable. The spectrum of the linear combination must be all real, too [Warming75], [Chen12].
To facilitate the analysis, we chose to use the non-conservative version of the governing equations. Define the non-conservative variables
(12)
Aided by using (12) and (5), the transformation between the conservative variables and the non-conservative variables can be done with the transformation Jacobian defined as
(13)
Aided by writing (5) as
the inverse matrix of can be obtained
(14)
and .
The transformation matrix can be used to cast the conservative equations, (9), into non-conservative ones. Pre-multiplying to (9) gives
(15)
where
(16)
To help obtaining the expression of , and , we substitute (5) into (11) and get
(17)
The Jacobian matrices in (16) can be spelled out with the expressions in (13), (14), and (17)
(18)
(15) is hyperbolic where the linear combination of its Jacobian matrices , , and
(19)
where , and are real and bounded.
The linearly combined Jacobian matrix can be used to rewrite the three-dimensional governing equation, (15), into one-dimensional space
(20)
where
(21)
and aided by (19) and the chain rule
The eigenvalues of can be found by solving the polynomial equation for , and are
(22)
where , and . The corresponding eigenvector matrix is
(23)
The left eigenvector matrix is
(24)
Riemann Invariants¶
Aided by (23) and (24), can be diagonalized as
(25)
(25) defines the eigenvalue matrix of . Aach element in the diagonal of the eigenvalue matrix is the eigenvalue listed in (22). Pre-multiplying (20) with gives
Define the characteristic variables
(26)
such that
Then aided by the chain rule, we can write
(27)
The components of defined in (26) are the Riemann invariants.