A kind of direct numerical simulation method suitable for supercritical carbon dioxide jet flow has been discussed in this paper. The form of dimensionless nonconservative compressible Navier-Stokes equations in a two-dimensional cartesian coordinate system is derived in detail. High accurate finite difference compact schemes based on non-uniform grid system are introduced to solve the equations. The simulation results of the three vortex pairing phenomenon of plane mixing layer and a compressible axisymmetric jet flow field show that the discussed numerical simulation method is feasible to calculate the supercritical carbon dioxide jet fluid. And it is found that the difficulties of splitting the convective terms in conservation Navier-Stokes equations, which are brought by the supercritical carbon dioxide fluid pressure state equation, can be avoided by solving the nonconservative compressible Navier-Stokes equations.
Supercritical carbon dioxide fluid refers to carbon dioxide fluid at more than the critical temperature and critical pressure. Its density is close to liquid, its diffusivity and viscosity is close to gas, and it has strong solvency. In recent years, much attention has been paid to the application of supercritical CO2 jet fluid technology in the petroleum engineering [
So far, there are two kinds of numerical simulation methods for supercritical co2 flow both at home and abroad. One is direct numerical simulation method, the other is turbulence model numerical simulation method. Direct numerical simulation is to solve directly the Navier-Stokes equations without introducing any closed models in the scales of turbulence grid size. At present, there are seldom research results on the supercritical co2 flow direct numerical simulation results, in which the representative results are as following. In 2006 Ri Shinryo et al. [
Based on the above content, a kind of direct numerical simulation method suitable for supercritical carbon dioxide jet flow is discussed in this paper. Supercritical carbon dioxide jet flow belongs to compressible flow problems. Navier-Stokes equations for compressible fluid flow can be expressed into two forms: conservation and nonconservation. In the general theory of fluid mechanics, it is same in nature to use the conservation form or the nonconservation form of Navier-Stokes equations, because it can be derived from one form to another form through simple derivation. However, which form of Navier-Stokes equations would be used is very important in computational fluid mechanics [
The conservative compressible Naver-Stokes equations are usually used to simulate compressible flows for a perfect gas. Because experience shows that the calculated flow field is generally smooth and stable using the conservative equations. In some literatures [7,8], a class of effective numerical simulation methods are proposed for conservative compressible Navier-Stokes equations. For example, In literature [
It is usually difficult to obtain a pressure state equation just as described in literature [
For convenience, the form of dimensionless nonconservative compressible Navier-Stokes equations in a twodimensional cartesian coordinate system is only presented in this paper.
The detail derivation process of nonconservative compressible Navier - Stokes equations has been showed in literature [
In equations (1), u and v, respectively, is corresponding to the speed in x and y directional coordinate. ρ and T, respectively, is corresponding to the density and the temperature. t is corresponding to the time. The pressure p is determined by the pressure state equation. The heat transfer coefficient is defined as
.
μ is viscosity, Cp is specific heat at constant pressure, Pr is a dimensionless number named as Prandtl number. In equations (1), there is
,.
is internal energy. In equations (1), , , and, respectively, is the weight of the viscous stress tensor, which is expressed as following:
,
and
.
In the thermal equilibrium state system, the internal energy has the general form as. It is known that the pharmaceutical units of Cp is
and the pharmaceutical units of T is K. So the pharmaceutical units of is m2/s2 just as similar as that of. According to the homogeneity principle of dimension, internal energy e can be defined as. So another form of equations (1) can be obtained as following:
Equations (2) should be changed into dimensionless form in order to be used to simulate some flow fields. The dimensionless system is chosen here as following:
, , ,
, , ,
, , ,
, , ,
and.
Here the physical quantities with a subscript “0” are corresponding characteristic quantities; and the physical quantities with a subscript “*” are corresponding dimensionless quantities. Thus the equations (2) can be written as following:
In equations (3), the characteristics of time can be instead of
and the characteristics of viscous stress can be instead of
.
So the following equations can be obtained as following:
(4)
In equations (4), the Reynolds number is defined as
.
According to the definitions of Mach number as
and speed of sound as
the definition of the pressure characteristics can obtained as
.
So there are
and
in equations (4). According to the expression of
and
in equations (1), the expression of
and
can be obtained. So there is
in equations (4).
can be defined in equations (4). It is known that the pharmaceutical units of pressure is, the pharmaceutical units of density is, the pharmaceutical units of specific heat at constant pressure is
and the pharmaceutical units of temperature is K, so it can be found that Sa is also a dimensionless number.
Based on the above analysis, the form of dimensionless nonconservative compressible Navier-Stokes equations in a two-dimensional cartesian coordinate system can be presented as following:
In equations (5), there are some expressions as following:
;;;
;;;;
;
.
In order to obtain better numerical simulation results, high accurate compact finite difference schemes can be used to discrete equations (5). Because non-uniform grids ussually need to be used for numerical simulation of complex flow field, a kind of high accuracy difference scheme based on non-uniform grid system can be used [
The one order spatial derivatives on the left of equations (5), such as
have the nature of the hyperbolic equation. So the basic characteristics of the disturbance wave propagation should be considered to using the upwind difference scheme to discrete the one order spatial derivatives on the left of equations (5). For example, the frozen coefficient method can be used to deal with the term of
, i.e.
.
Because the coefficient
the 5th order upwind compact backward difference scheme based on non-uniform grid system can be used to discrete the term
, i.e.
Using Taylor series expansion, the coefficients as, , , and in equation (6) can be solved through the algebraic equations, i.e.
Because the coefficient
the 5th order upwind compact forward difference scheme based on non-uniform grid system can be used to discrete the term
, i.e.
Using Taylor series expansion, the coefficients as, , , and in equation (8) can be solved through the algebraic equations, i.e.
The 6th order symmetric compact difference scheme based on non-uniform grid system can be used to discrete the spatial derivative terms on the right in the equations (5). For example, the term of
can be dealed with as following:
Using Taylor series expansion, the coefficients as, , , , and in equation (10) can be solved through the algebraic equations, i.e.
The 3th order accurate Rungge-Kutta method can be used to discrete the time terms on the left in the equations (5) [
In order to verify the above numerical simulation method for the non conservative compressible Navier-Stokes equations, two numerical simulation experiments are analyzed in this paper.
The three vortex pairing phenomenon of plane mixing layer under the condition of flow Maher number and Reynolds number was simulated numerically using the above method.
The initial flow field is the mean flow field adding the turbulence flow field as, here. The mean flow field is give as:, ,
,.
The turbulence flow field is given as [
Here, means the most unstable wave number which is obtained from the linear theory;; the characteristic functions as and are obtained from numerical linear stability analysis; the turbulence amplitude is given as;; the phase difference.
The periodic boundary is used in the x direction, and the no reflection boundary is used in the y direction. The computational domain is given as and. The uniform grid is used in the x direction. The nonuniform grid is used in the y direction, and the grid is densed near. The grid numbers are.
A compressible axisymmetric jet flow field was simulated numerically using the above method to slove the compressible nonconservative two-dimensional Navier - Stokes equations in the column coordinates, as the incoming flow Mach number is and the Reynolds number is. The initial conditions are given as:
, ,
, ,.
The no reflection boundary condition is used on the upper and export boundary of the flow field. According to physical characteristics of the flow, symmetrical condition is adopted for u, ρ and T, and the antisymmetric condition is adopted for v on the jet symmetric axis. The computational domain is given as,. Uniform grids are adopted in the x direction, and non-uniform grid is adopted in the r direction with local mesh encryption near the axis of symmetry. The grid number is.
A kind of direct numerical simulation method suitable for supercritical carbon dioxide jet flow has been discussed in this paper. Firstly, the form of dimensionless nonconservative compressible Navier-Stokes equations in a two-dimensional cartesian coordinate system is de-
rived in detail. High accurate finite difference compact schemes based on non-uniform grid system are introduced to solve the equations. Then, three vortex pairing phenomenon of plane mixing layer and a compressible axisymmetric jet flow field are simulated using the numerical method proposed in this paper. The relevant results show that the above numerical simulation method of solving dimensionless nonconservative compressible Navier-Stokes equations is feasible to calculate the supercritical carbon dioxide jet flow. And it is found that
the difficulties of splitting the convective terms in conservation Navier-Stokes equations, which is brought by the supercritical carbon dioxide fluid pressure state equation, can be avoided by solving the nonconservative compressible Navier-Stokes equations.
The preliminary discussion about a direct numerical simulation method for supercritical carbon dioxide jet flow only has been performed in this paper. In the later research, the supercritical CO2 jet flow structure would be presented in detail.