Crank-Nicholson Scheme of the Zeroth-Order Approximate Deconvolution Model of Turbulence Based On a Mixed Formulation

(explained in Section 2), the ADM for Navier-Stokes equations is, with the new model variables (w, q) , follows: Abstract This report presents a method with high spatial and temporal accuracy for es-timating solutions of Navier-Stokes equations at high Reynolds number. It em-ploys Crank-Nicolson time discretization along with the zeroth-order approximate deconvolution model of turbulence to regularize the flow problem; solves a deviation of the Navier Stokes equation instead. Both theoretical and computational findings of this report illustrate that the model produces a high order of accuracy and stability. Furthermore, measurements of the drag and lift coefficients of a benchmark problem verify the potential of the model in this kind of computations.


Introduction
Navier-Stokes equations represent the motion of the fluid flow. In the incompressible flow case, it seeks for the velocity-pressure pair : Ω × [0, ] → ( = 2,3) and : Ω × (0, ] → satisfying + ⋅ ∇ − Δ + ∇ = , for ∈ Ω, 0 < ≤ , ∇ ⋅ = 0, ∈ Ω, for 0 ≤ ≤ , for 0 < ≤ . Throughout this paper, for theoretical reasoning, we consider the case of periodic boundary conditions. Nevertheless, numerical experiments with Dirichlet boundary conditions still performs as expected. Its well-known that fluid flow is continuum of scales, where the smallest persistent scales in the case of a 3D flow is of the order ( −3/4 ) in size where the Reynolds number is inverse proportional to the viscosity coefficient . Considering real-life applications of this equation, e could easily be around ~(10 10 ) and even more for large domains, e.g., for atmosphere-ocean interaction. Capturing all persistent scales, in many applications is not feasible or indeed impossible. Contrarily, the large scales in the flow are responsible for much of the mixing and most of the momentum transport. Upon this observation, turbulence models like Large Eddy Simulation (LES) gained much attention [1,2,3]. The idea is that set a filter width > 0 for the size of large scales and model the effect of the smaller scales to these large scales. This merely requires solving for large scales (bigger than ), and hence avoids singularities due to small turbulent fluctuations.
An LES model is required when one tries to close the system (2) by choosing an approximation to the last term on the left-hand side of (2a). In this report, we consider the family of Approximate Deconvolution Models (ADMs). These models were introduced by Stolz and Adams in [4] and further studied in e.g., [5][6][7][8][9][10]. Given the family of approximate deconvolution operators (explained in Section 2), the ADM for Navier-Stokes equations is, with the new model variables (w, q), as follows:

Abstract
This report presents a method with high spatial and temporal accuracy for estimating solutions of Navier-Stokes equations at high Reynolds number. It employs Crank-Nicolson time discretization along with the zeroth-order approximate deconvolution model of turbulence to regularize the flow problem; solves a deviation of the Navier Stokes equation instead. Both theoretical and computational findings of this report illustrate that the model produces a high order of accuracy and stability. Furthermore, measurements of the drag and lift coefficients of a benchmark problem verify the potential of the model in this kind of computations.
The spaces and are defined in Section 2. Zeroth-order ADM attains high spatial accuracy. However, time accuracy is also crucial for turbulence modeling. Accordingly, it shall be supplied with a stable high accuracy time discretization. This goal has been addressed with a predictor-corrector type deferred correction method in [11] and [12]. In this report, we investigate Crank-Nicolson scheme employed for zeroth-order ADM (CN-ADM).
The report is organized as follows. In Section 2, we introduce the necessary notations and preliminary results. The new method is introduced, and the stability and accuracy results are given in section 3. The numerical tests involving a quantitative and a qualitative assessment will be presented in Section 4.
The space that the velocity (at time ) will be sought in is given by We assume throughout the paper that the velocitypressure finite element spaces ℎ ⊂ and ℎ ⊂ are conforming, have common approximation properties of finite element spaces, and also satisfy the discrete inf-sup, or ℎ , condition where ℎ is bounded away from zero uniformly in ℎ.

Definition 1 (Approximate Deconvolution Operator)
For a fixed finite , define the ℎ approximate deconvolution operator by where the averaging operator −1 is the differential filter: given ∈ 2 (Ω), ∈ 2 (Ω) is the unique solution of under periodic boundary conditions. With periodicity assumption on boundaries, this operator commutes with differentiation.

Lemma 2
The operator is compact, positive, and is an asymptotic inverse to the filter −1 , i.e., for very smooth and as → 0, it satisfies The proof of the lemma can be found in [10]. We also define the following norm, induced by the deconvolution operator : In the analysis we use the properties of the following Modified Stokes Projection (see [6]).
To that end, we assume the mesh to be quasi-uniform.

Preposition 4 (Stability of the Stokes Projection)
Let ̃, ̃ satisfy (9) for given , . The following bound holds: In the error analysis of the proposed method, we use the error estimate of the Stokes projection (9).
We will also need the following inequalities: for any The proof of (10) can be found in [13], and (11) follows from the Poincare-Friedrich's inequality and (10).

2Δ
(| ℎ, +1 | − | ℎ, | ) Using the definition of −1 (Ω)-norm on the right-hand side and applying Young's inequality Multiplying both sides by 2Δ and summing over all time levels gives the desired result.
Use the approximation properties of ℎ , ℎ . Since the mesh nodes do not depend upon the time level, it follows from (10) and (11) that Under the assumption of the theorem Applying the discrete Gronwall's lemma and using bounds in the stability and the accuracy of the modified stokes projection theorem, give It must be noted that initial || ℎ,0 || 2 + 2 ||∇ ℎ,0 || 2 errors stays within the desired accuracy due to the preposition 4 of the modified Stokes projection.
The stated error estimate now follows by applying the triangle inequality.

Corollary 9 (Optimal Convergence Rate)
Suppose that in addition to the assumptions of the above theorem, ℎ and ℎ are composed of Taylor-Hood finite elements (P2 − P1). Then the error is of the form, || − ℎ, ||

Numerical Results
First, we perform a convergence test for the proposed CN-ADM. In this test non-homogeneous Dirichlet boundary conditions are implemented. Although theory of the model requires periodicity, this choice of boundary conditions still performs as Corollary (9)  The forcing and initial condition 0 are computed so that it complies with (1). The final time in the computations is = 1. In order to verify the theoretical claims on the convergence rates, we take Δ = ℎ = 2 = 2 − . For = 0.1 the convergence rates in Table 1 confirm what is predicted by Corollary (9): the convergence rates of 2 nd order are achieved. Also, we observe the asymptotic character of convergence, which is typical for the ADM methods, [8,9,11,12].
For the qualitative assessment, consider the 2D flow past an obstacle Figure 1, see [14]- [16]. In this setting the channel height is = 0.41 , the channel length is = 2.2 , the diameter of the circle whose center is located at 0.2 above the lower wall and 0.2 right of the left wall of the channel is = 0.1 .   Choosing ν = 0.001, this construction results in a time dependent Reynolds number, 0 ≤ ( ) ≤ 100.
The correct behavior for this setting is, from time t = 2s to t = 4s, two vortices start to develop behind the obstacle, and then they separate, later on a vortex street forms which can be visible until the final time t = 8, see [15], [16]. Plots of the flow at t = 2,4,6 and 8 are shown in Figure 2. These results look comparable with their corresponding plots in [15]- [16].
We also compute the maximum drag (C D ) and lift (C L ) coefficients along with the pressure differences as in [15] with Taylor-Hood elements. The filter radius δ is chosen to be the average mesh width. It must be noted that the proposed CN-ADM algorithm is a turbulence model; targeted to solve a variation of NSE not the NSE itself. Also, our computations are performed on a very coarse mesh with a higher time step-size. Therefore, an exact match with the reference values of lift and drag is not required. Nevertheless, we should observe results close by these values to consider CN-ADM useful. In this regard, we mostly compare our results with those obtained in [16]. Table 3 suggests that CN-ADM performs quite better than the Leray model (without any indicator given in [16]) solved with 14,446 degrees of freedom. Comparing to the finer-mesh results with Leray model, CN-ADM seems to produce better results for lift coefficient even if it was computed 10 times coarse time step size and around 3.4 times less degrees of freedom. For the drag coefficient and pressure drop, Leray model with higher degrees of freedom produces better results.

Conclusions
A turbulence model with high accuracy both in temporal and spatial variables presented in this report. Both theory and computational results suggest that proposed high accuracy is indeed achieved. Also, a qualitative testing is presented that match with its reference turbulent behavior. In addition to these, computations for drag and lift coefficients of a benchmark problem is presented: illustrating this model could be employed for such computations beside any qualitative assessments.