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This paper describes a characteristics-mix finite element method for the computation of incompressible Navi-er-Stokes equations with variable density. We have introduced a mixed scheme which combines a characteristics finite element scheme for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint. The proposed method has a lot of attractive computational properties: parameter-free, very flexible, and averting the difficulties caused by the original equations. The stability of the method is proved. Finally, several numerical experiments are given to show that this method is efficient for variable density incompressible flows problem.

This paper is devoted to the numerical approximation of incompressible viscous flows with variable density. This type of flows is governed by the time-dependent Navier-Stokes equations [

where the dependent variables are the density

where

Compared with the constant density incompressible Navier-Stokes equation, the main difficulty for the simu- lation of the system Equation (1)-(2) is that these equations entangle hyperbolic, parabolic, and elliptic features. Therefore, how to construct stable and efficient numerical schemes for the system Equations (1) and (2) is challenging.

For developing numerical approximations to this problem, it seems natural to use, as far as possible, the techniques established for the solution of constant density incompressible Navier-Stokes equations, viz., the fractional step projection method of Chorin [

In this paper, we consider the conserved form for variable density incompressible flows which is introduced by Guermond et al. [

In the above equations Equation (3), the additional term

zero if

The originality of our work is to use different numerical methods for the transport equation and for evaluating the evolution of the velocity driven by the last two equations in the system Equation (1). To be more specific, we use a time-splitting, solving the first equation for a given velocity by using a characteristic stabilized finite element approach which is efficient when dealing with a pure convection equation, and then, we compute the divergence free solution of the last two equations by exploiting the advantages of FE methods, see [

The paper is organized as follows. In next section, we introduce some notations for this paper. In Section 3, a detailed presentation of the new method is given. In Section 4, the stability of the method is proved. In Section 5, a series of numerical experiments are given. The last section is devoted to concluding remarks.

In this section, we aim to describe some of the notations which will be frequently used in this paper. We con- sider the time-dependent variable density Navier-Stokes system Equations (1) and (2) on the finite time interval

Let

No notational distinction is done between scalar or vector-valued functions but spaces of vector-valued functions are identified with bold fonts. The space of functions in

For the mathematical setting of problem Equation (1), we introduce the following Hilbert spaces:

The spaces

We also introduce the following bilinear operator:

and a trilinear form on

Obviously, the bilinear

where

Henceforth

Set

Step 1. Solve new density field:

Find

Step 2. Solve new velocity and pressure fields:

Find

The origin of our scheme can be seen by considering the first equation in a space-time framework. First, for density field

where

Let us introduce the characteristic curves, which are simply the the trajectories of the motion associated with the velocity field

which can be obtained by solving the initial value problem

It represents the trajectory described by a material point that is placed at position

By using function

For the time variable, the solution will be approximated at times

In order to discretize the material time derivative in Equation (5), we use the following first order backward Euler formula, namely,

Moreover, for

So, let us introduce the following characteristics scheme for time semidiscretization of problem Equation (5)

Now, multiplying Equation (14) by a test function

Find

We define finite element space

where

Find

In the numerical simulation of the Navier-Stokes Equation (6), a major difficulty is that the velocity and the pressure are coupled by the incompressibility constraint. Many researchers have done a lot of work about Navier- Stokes system with constant density, for example [

Since we aim at using a FE method, it is convenient to write the variational formulation of Equation (6). Let

Find

The domain

FE spaces

of spaces

where for all

Find

In the next section, we will prove that the above algorithm is stable.

In this section, we recalls some useful Propositions and stability hypothesis assumptions for the characteristics- mix finite element method for the incompressible flow with variational density [

Moreover, we also need the following results [

Propositon 4.1. For

Next, we start with the stability proof of the system. To avoid irrelevant technicalities, we assume that there is no external driving force, i.e.,

Lemma 4.2. Let

where

Proof: Taking

Using (4.1), implies that

Substituting above inequality into (4.4) and using (4.1), yields

So, we have

Similarly,

Substituting above inequality into (4.4) again, leads to

So, we have

which together with (4.6), we can easily obtain the following result

Next, set

and using the (4.8), a simple derivation leads to

Finally, we obtain the desired stability result.

Theorem 4.3. Let

where

Proof: Taking

Using (4.1), implies that

Substituting the above inequality into (4.11) and using (4.1), yields

So, we have

Then using the Poincare inequality, we obtain

Therefore, we can easily get

Next, using the above inequality and Poincare inequality again, we have

Also, combining (4.14), (4.15) with (4.9), we obtain the desired stability result.

In this section, we present four series of numerical results to illustrate the theoretical analysis of the algorithm proposed in this paper.

In order to test the accuracy of the algorithm proposed in this paper, we consider a problem with a known analy- tical solution. We solve the variable density Navier-Stokes equations Equations (1) and (2) in the unit square

so that the right-hand side to the momentum equation is

We use the

First, we solve the above mentioned problem for

Secondly, computation are made on a fixed mesh size for different Reynolds number (Re = 1000, 3000, 5000,

8000, 10000). Taking

we can see that the stability still keeps well when the Reynolds number increases. These demonstrate that our method is very effective for high Reynolds number.

Next, computation are made on a fixed mesh size and a fixed Reynolds number with different time steps. The computation has been performed for

In this Subsection we illustrate the performance of the method on a realistic problem, namely we investigate a Rayleigh-Taylor instability. The problem has been considered in [

Order | Order | Order | Order | |||||
---|---|---|---|---|---|---|---|---|

8 | 1.13557e−4 | 1.52491e−4 | 4.40802e−3 | 4.63947e−3 | ||||

16 | 2.13e−5 | 2.4145 | 2.58284e−5 | 2.5617 | 1.46433e−3 | 1.5899 | 1.12825e−3 | 2.0399 |

24 | 6.23631e−6 | 3.0294 | 7.29882e−6 | 3.1168 | 6.19744e−4 | 2.1206 | 4.84383e−4 | 2.0854 |

32 | 2.43546e−6 | 3.2684 | 2.79238e−6 | 3.3399 | 3.06827e−4 | 2.4437 | 2.65326e−4 | 2.0923 |

Order | Order | Order | Order | |||||
---|---|---|---|---|---|---|---|---|

0.1 | 3.9601e−3 | 1.7513e−2 | 0.45137 | 0.118272 | ||||

0.05 | 2.2643e−3 | 0.8065 | 8.7673e−3 | 0.9982 | 0.224569 | 1.0072 | 0.0530867 | 1.1557 |

0.025 | 1.3818e−3 | 0.7125 | 4.3543e−3 | 1.0097 | 0.110574 | 1.0221 | 0.0244584 | 1.1180 |

0.0125 | 8.0274e−4 | 0.7836 | 2.1559e−3 | 1.0141 | 0.054659 | 1.0165 | 0.0115745 | 1.0794 |

0.00625 | 3.5149e−4 | 1.1914 | 1.0798e−3 | 0.9975 | 0.027380 | 0.9973 | 0.0059369 | 0.9632 |

0.003125 | 1.6324e−4 | 1.1065 | 5.5062e−4 | 0.9717 | 0.013947 | 0.9732 | 0.0030605 | 0.9560 |

which splits into two region with varying density, the heavier fluid superposed to the light one. The interface is slightly smoothed since we set at time

with

1) the density ratio between the light and the heavy fluid, which is measured by the so-called Atwood number

2) the Reynolds number, defined as

where

The equations are made dimensionless by using the following references:

Next, we compare the solutions obtained at different Atwood numbers.

・ A low Atwood number problem: Setting

・ A high Atwood number problem: Setting

・ A very high Atwood number problem: Setting

To investigate the capability of our method to work with larger density variations, we give the computational results for rising bubble test. This simulation is inspired from [

where

ing references:

The results are displayed in

To investigate the capability of our method to work with very large density variations, a two-fluid flow in a sloshing tank is considered next. The setup of the test case follows the description [

The densities of the fluids are

is considered here, so the volume force is

the walls of the tank, and a zero velocity field is initially assumed. The time-step length is

The results are displayed in

In this paper, we proposed a characteristics-mix finite element method to the case of incompressible viscous flows with variable density. The originality of our approach is to use different numerical methods for the transport equation and evaluating the evolution of the velocity pressure. The new method uses a time splitting, solving separately the transport equation and the momentum equation. To be more specific, we solve the first equation for a given velocity by using a characteristic stabilized finite element approach which is efficient when dealing with a pure convection equation, and then, we compute the divergence free solution of the last two equations by exploiting the advantages of FE methods. The stability proof of the method we proposed for variable density flows was given in the paper.

To verify the correctness of the method, it has been applied to the test cases previously considered in the literature. The spatial approximation is performed by means of Lagrangian finite elements with P2 interpolation for density and velocity and P1 interpolation for pressure. First, the rates of convergence of the method were proved to be in accordance with the theoretical expected ones, leading so to an accurate solver. Then, the simulation of the viscous Rayleigh-Taylor instability was also investigated. We obtained very good results, even for rather high Atwood numbers. Finally, we considered the rising bubble test and sloshing tank to investigate the robustness property of the scheme with regard to high density ratios. The simulation results coincided with the law of physics are very close to the results presented in the literature. Compared with some established methods, the numerical results show that the new method exhibits good stability behavior even large time steps or the high Atwood numbers are used in computation.