`Advances in Numerical AnalysisVolumeÂ 2010Â (2010), Article IDÂ 419021, 21 pageshttp://dx.doi.org/10.1155/2010/419021`
Research Article

Discontinuous Time Relaxation Method for the Time-Dependent Navier-Stokes Equations

Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA

Received 17 July 2010; Accepted 16 September 2010

Copyright Â© 2010 Monika Neda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A high-order family of time relaxation models based on approximate deconvolution is considered. A fully discrete scheme using discontinuous finite elements is proposed and analyzed. Optimal velocity error estimates are derived. The dependence of these estimates with respect to the Reynolds number Re is , which is an improvement with respect to the continuous finite element method where the dependence is .

1. Introduction

Turbulence is a phenomenon that appears in many processes in the nature, and it is connected with many industrial applications. Based on the Kolmogorov theory [1], Direct Numerical Simulation (DNS) of turbulent flow, where all the scales/structures are captured, requires the number of mesh points in space per each time step to be in three-dimensional problems, where is the Reynolds number. This is not computationally economical and sometimes not even feasible. One approach is to regularize the flow and one such type of regularization is the time relaxation method, where an additional term, the so-called time relaxation term, is added to the Navier-Stokes equations (cf. Adams and Stolz [2] and Layton and Neda [3]). The contribution to the Navier-Stokes equations from the time relaxation term induces an action on the small scales of the flow in which these scales are driven to zero. This time relaxation term is based on filtering and deconvolution methodology. In general, many spacial filtering operators associated with a length-scale are possible (cf. Berselli et al. [4], John [5], Geurts [6], Sagaut [7] and Germano [8]). First, consider the equations of differential filter (cf. Germano [8]) where , is the domain and is its boundary. Here represents the averaging radius, in general, chosen to be of the order of the mesh size.

The deconvolution algorithm that it is considered herein was studied by van Cittert in 1931, and its use in Large Eddy Simulation (LES) pioneered by Stolz and Adams (cf. Stolz and Adams [2, 9]). For each it computes an approximate solution by N steps of a fixed point iteration for the fixed point problem (cf. Bertero and Boccacci [10]) The deconvolution approximation is then computed as follows.

Algorithm 1.1 (van Cittert approximate deconvolution algorithm). Consider that , for , perform

By eliminating the intermediate steps, it is easy to find an explicit formula for the th deconvolution operator given by For example, the approximate deconvolution operator corresponding to is

Many fluid models that are based on numerical regularization and computational stabilizations have been explored in computational fluid dynamics. One such regularization and the most recent has been proposed by Stolz et al. [11, 12] and arises by adding a linear, lower order time regularization term, , to the Navier-Stokes equations (NSE). This term involves which represents the part of the velocity that fluctuates on scales less than order and it is added to the NSE with the aim of driving the unresolved fluctuations of the velocity field to zero. The time relaxation family of models, under the no-slip boundary condition, is then defined by where , is a convex bounded regular domain with boundary , is the fluid velocity, is the fluid pressure and is the body force driving the flow. The kinematic viscosity is inversely proportional to the Reynolds number of the flow. The initial velocity is given by . A pressure normalization condition is also needed for uniqueness of the pressure. The time relaxation coefficient has units . The domain is two-dimensional, but the numerical methods and the analysis can be generalized to three-dimensional domains, as stated in [13] for the case of Stokes and Navier-Stokes problems.

Existence, uniqueness and regularity of strong solutions of these models are discussed in [3]. Even though there are papers on the simulation of the models for incompressible and compressible flows, there is little published work in the literature on the numerical analysis of the models. In [14], a fully discrete scheme using continuous finite elements and Crank-Nicolson for time discretization is analyzed and the energy cascade and joint helicity-energy cascades are studied in [3, 15], respectively.

In this work, a class of discontinuous finite element methods for solving high-order time relaxation family of fluid models (1.7)â€“(1.10) is formulated and analyzed. The approximations of the averaged velocity and pressure are discontinuous piecewise polynomials of degree and , respectively. Because of the lack of continuity constraint between elements, the Discontinuous Galerkin (DG) methods offer several advantages over the classical continuous finite element methods: (i) local mesh refinement and derefinement are easily implemented (several hanging nodes per edge are allowed); (ii) the incompressibility condition is satisfied locally on each mesh element; (iii) unstructured meshes and domains with complicated geometries are easily handled. In the case of DNS, DG methods have been applied to the steady-state NSE (cf. Girault et al. [16]) and to the time-dependent NSE (cf. Girault et al. [17]) where they are combined with an operator-splitting technique. Another discontinuous Galerkin method for the NSE based on a mixed formulation are considered in [18] by Cockburn et al.. For high Reynolds numbers, the numerical analysis of a DG scheme combined with a large eddy simulation turbulence model (subgrid eddy viscosity model) is derived in [19] by Kaya and RiviĂ¨re.

This paper is organized in the following way. Section 2 introduces some notation and mathematical properties. In Section 3, the fully discrete schemes are introduced and it is proved that the schemes solutions are computable. A priori velocity error estimates are derived in Section 4. The family of models (1.7)â€“(1.10) is regularization of the NSE. Thus, the correct question is to study convergence of discretizations of (1.7)â€“(1.10) to solutions of the NSE as and (rather than to solution of (1.7)â€“(1.10)). This is the problem studied herein. Conclusions are given in the last section.

2. Notation and Mathematical Preliminaries

To obtain a discretization of the model a regular family of triangulations of , consisting of triangles of maximum diameter , is introduced. Let denote the diameter of a triangle and the diameter of its inscribed circle. Regulary, it is meant that there exists a parameter , independent of , such that This assumption will be used throughout this work. denotes the set of all interior edges of . Let denote a segment of shared by two triangles and () of ; it is associated with a specific unit normal vector directed from to and the jump and average of a function on is formally defined by If belongs to the boundary , then is the unit normal exterior to and the jump and the average of on coincide with the trace of on .

Here, for any domain , is the classical space of square-integrable functions with inner-product and norm . The space is the subspace of functions of with zero mean value The standard Sobolev spaces are denoted by where is the , with norm and seminorm .

Next, the discrete velocity and pressure spaces are defined to be consisting of discontinuous piecewise polynomials. For any positive integer , the corresponding finite-dimensional spaces are where is defined as the span of polynomials of order over triangle .

Denoting by the measure of , the following norms are associated for the spaces and where is the broken norm defined by

Finally, some trace and inverse inequalities are recalled, that hold true on each element in , with diameter , the constant is independent of

3. Numerical Methods

In this section, the DG scheme is introduced and the existence of the numerical solution is shown. First, the bilinear forms are defined and by

The parameter takes the value or : this will yield different schemes that are slight variations of each other. It will be showed that all the resulting schemes are convergent with optimal convergence rate in the energy norm . In the case where , the bilinear form is symmetric; otherwise it is nonsymmetric. We remark that the form is the standard primal DG discretization of the operator . Finally, if is either or , the jump parameter should be chosen sufficiently large to obtain coercivity of (see Lemma 3.1). If , then the jump parameter is taken equal to .

The incompressibility condition (1.8) is enforced by means of the bilinear form defined by Finally, the DG discretization of the nonlinear convection term , which was introduced in [16] by Girault et al. and studied extensively in [16, 17] by the same authors, is recalled as follows: where the superscript denotes the dependence of on and the superscript (resp., ) refers to the trace of the function on a side of coming from the interior of (resp., coming from the exterior of on that side). When the side of belongs to , the convention is the same as for defining jumps and average, that is, the jump and average coincide with the trace of the function. Note that the form c is not linear with respect to , but linear with respect to and .

Some important properties satisfied by the forms (cf. Wheeler [20], and Girault et al. [16, 17]) are recalled.

Lemma 3.1 (Coercivity). If , assume that . If , assume that is sufficiently large. Then, there is a constant , independent of , such that

It is clear that if . Otherwise, is a constant that depends on the polynomial degree of and of the smallest angle in the mesh. A precise lower bound for is given in [21] by Epshteyn and RiviĂ¨re.

Lemma 3.2 (Inf-sup condition). There exists a positive constant , independent of such that

Lemma 3.3 (Positivity). One has

The discrete form of the differential filter (1.1) is defined following the work of Manica and Merdan [22].

Definition 3.4 (Discrete differential filter). Given , for a given filtering radius , where is the unique solution in of

Remark 3.5. An attractive alternative is to define the differential filter by a discrete Stokes problem so as to preserve incompressibility approximately [23â€“25].

Definition 3.6. The discrete van Cittert deconvolution operators are where is the projection.

For , the discrete deconvolution operator for is

was shown to be an approximate inverse to the filter operator in Lemma of Dunca and Epshteyn [26], recalled next.

Lemma 3.7. is a bounded, self-adjoint positive operator. is an asymptotic inverse to the filter . Specifically, for smooth and as ,

Some basic facts about discrete differential filters and deconvolution operators are presented next.

Lemma 3.8. For , one has the following bounds for the discretely filtered and approximately deconvolved :

Proof. The proof of (3.12) follows from the standard finite element techniques applied on the discretized equation (3.8) of the filter problem (1.1). Pick . Then, using coercivity result The term is positive, so it will be dropped, which yields Multiplying by and taking the square root yields the estimate (3.12). Equation (3.13) follows immediately from (3.12) and the definition of .

Lemma 3.9. For smooth the discrete approximate deconvolution operator satisfies

Proof. The proof follows the same arguments as in [27] for the case of continuous finite element discretization of the filter problem. The error is decomposed in the following way: Lemma 3.7 gives The standard discontinuous finite element bound for (3.8) is given by (cf. RiviĂ¨re [13])
Lemma 3.8 gives for the third term in (3.17) that . Then, (3.19) is applied.
Now, it is left to bound the second term from (3.17). First, note that for , . Based on Definition 3.6 of continuous and discrete deconvolution operators and their expansion (see (3.10)), is a polynomial of degree in (and in as well). Thus, the second term in (3.17) can be written as
For coefficients and for , the result (3.19) gives
For , the results (3.19) and (3.12) give
Inductively,
The proof is completed by combining the derived bounds for the terms in (3.17).

Remark 3.10. There remains the question of uniform in bound of the last term, , in (3.16). This is a question about uniformregularity of an elliptic-elliptic singular perturbation problem and some results are proven in [28] by Layton. To summarize, in the periodic case it is very easy to show by Fourier series that for all The nonperiodic case can be more delicate. Suppose and on (i.e., ). Call so satisfies Then it is known that , and on . Further, So, (3.24) holds for . It also holds for higher values of provided additionally on for .
Now consider the second term , that is, . We know from elliptic theory for , that , (as noted above) on and Theorem in [28] then implies, uniformly in , This extends directly to .

Extending Lemma 3.9, the following assumption will be made.

Assumption DG1. The terms in (3.16) are independent of and

The minimal conditions that are assumed throughout are that the (discrete) filter and (discrete) deconvolution used satisfy the following consistency conditions of Stanculescu [29].

Assumption DG2. and are symmetric, positive definite () operators.

These have been proven to hold for van Cittert deconvolution (cf. Stanculescu [29], Manica and Merdan [22] and Layton et al. [27]). For the DG method, the second assumption restricts our parameter for the discretization of the filter problem (3.8), so that the bilinear form is symmetric.

The numerical scheme that uses discontinuous finite elements in space and backward Euler in time is derived next. For this, let denote the time step such that is a positive integer. Let be a subdivision of the interval . The function evaluated at the time is denoted by . With the above forms, the fully discrete scheme is: find such that:

Remark 3.11. The time relaxation term can be treated explicitly such that the optimal accuracy and stability are obtained and this would make the scheme much easier to compute [30].

The consistency result of the semidiscrete scheme is showed next.

Lemma 3.12 (Consistency). Let be the solution to (1.7)â€“(1.10), then satisfies where the consistency error . Furthermore,

Proof. Equations (3.34) and (3.35) are clearly satisfied because of (1.8), (1.9), and (1.10) and the regularity of . Next, we multiply (1.7) by and integrate over one mesh element Summing over all elements
By Green's formula The regularity of then yields Note that Green's formula yields and that the incompressibility condition with the regularity of yields The final result is obtained by bounding the consistency error . For , we have For , The bound for is obtained by applying an induction argument and Remark 3.10.

The existence and uniqueness of the discrete solution is stated next.

Proposition 3.13. Assume that Lemma 3.1 holds. Then, there exists a unique solution to (3.30)â€“(3.32).

Proof. The existence of is trivial. Given , the problem of finding a unique satisfying (3.30)-(3.31) is linear and finite-dimensional. Therefore, it suffices to show uniqueness of the solution. Consider the problem restricted to the subspace defined by Let and be two solutions and let . Then, satisfies: Choosing and using the coercivity result (3.5), positivity result (3.7) and positivity of the operator given in Assumption DG2, we obtain: which yields that . The existence and uniqueness of the pressure is then obtained from the inf-sup condition (3.6).

Some approximation properties of the spaces and are recalled next. From Crouzeix and Raviart [31], and Girault et al. [16], for each integer , and for any , there is a unique discrete velocity such that Furthermore, if , there is a constant independent of such that

For the pressure space, we use the approximation given by the projection. For any , there exists a unique discrete pressure such that In addition, if , then

The discrete Gronwall's lemma plays an important role in the following analysis.

Lemma 3.14 (Discrete Gronwall's Lemma (cf. Heywood and Rannacher [32])). Let , and , , , (for integers be nonnegative numbers such that Suppose that , for all , and set . Then,

The family of time relaxation models is a regularization of NSE, and therefore it is natural to investigate the finite element error between the discretized model and the NSE. To that end, we will assume that the solution to the Navier-Stokes equations that is approximated is a strong solution and in particular satisfies (cf. RiviĂ¨re [13])

4. A Priori Error Estimates

In this section, convergence of the scheme (3.30)â€“(3.32) is proved. Optimal error estimates in the energy norm are obtained.

Theorem 4.1. Assume that , , , and . Assume also that the coercivity Lemma 3.1 holds and that satisfies (3.24). If is chosen of the order of , and , there exists a constant , independent of and but dependent on such that the following error bound holds, for any :

Remark 4.2. The dependence of these error estimates with respect to the Reynolds number Re ( 1/) is , which is an improvement with respect to the continuous finite element method where the dependence is .

Proof. Defining and subtracting (3.55) from (3.30), we have We now decompose the error , where and is the interpolation error . Choosing in the equation above, using the coercivity result (3.5) and positivity of the operator , we obtain for (4.2) Consider now the nonlinear terms from the above equation. First note that since is continuous, we can rewrite so, for readability, the superscript in the form is dropped. Therefore, adding and subtracting the interpolant yields Thus, the error equation (4.4) is rewritten as From property (3.7), the term in the left-hand side of (4.7) is positive and therefore it will be dropped. For the other terms of the form that appear on the right-hand side of the above error equation we obtain bounds, exactly as in the proof of Theorem in [19] by Kaya and RiviĂ¨re. The constant is a generic constant that is independent of and , and that takes different values at different places Therefore, we have
To bound , Cauchy-Schwarz's inequality, Young's inequality and the approximation result (3.50) for are applied To bound the term , a Taylor expansion with integral remainder is used This implies that Thus, with (3.50), we have Next, the term is expanded as The term is bounded using Cauchy-Schwarz inequality, Young's inequality and the approximation result (3.49) Using Cauchy-Schwarz's inequality, trace inequality (2.8) and approximation result (3.50), we have Using Cauchy-Schwarz's inequality, trace inequality (2.10), and approximation result (3.49), we have Using the approximation result (3.49), we have Putting together the bounds (4.15), (4.16), (4.17) and (4.18), For the terms and , Cauchy-Schwarz's inequality, Young's inequality and bounds (3.13) and (3.29) are applied Using (4.3) with (3.48) and (3.51), the pressure term is reduced to which is bounded by using Cauchy-Schwarz's inequality, Young's inequality, trace inequality (2.7) and approximation result (3.52) With the bounds (4.9), (4.10), (4.13), (4.19), (4.20), and (4.22), the error equation becomes where are constants independent of and . Now, multiply the equation by , sum from to , and use the assumption that is chosen of the order of to obtain Thus, using Gronwall's lemma with , there is a constant independent of and , but dependent on , such that The final result is then obtained by noting that the term is of order and by using triangle inequality and approximation result.

5. Conclusion

In this paper, a numerical scheme for solving the time relaxation family of models based on approximate deconvolution technique for fluid flow problems is formulated and analyzed. The proposed method is convergent with optimal convergence rates with respect to the mesh size. The approximations of the average velocity and pressure are discontinuous piecewise polynomials. One benefit of using discontinuous elements is that the error estimates depend on the Reynolds number as , whereas the dependence is for continuous finite elements (cf. [14]).

The proposed scheme (3.30)â€“(3.32) contains parameter in the bilinear form for the discretization of the viscous term that yields different numerical approximations. Only numerical simulations of benchmark problems for high Reynolds numbers, will help determine which choices of are preferred for a given mesh size. This is the object of a future paper.

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