Multi-resolution lattice Green’s function method for high Reynolds number external flows

Introduction

Motivation

In this work, we focus on the LGF method, which is described in detail in Section 2.2.2. Although the combination of LGF and IB methods is very effective (Liska and Colonius, 2017), it is still limited to moderate Reynolds numbers as they fail to resolve the scale separation associated with thin boundary layers and turbulence present in high Reynolds number flows .

Contributions and outline

In the second step, an AMR grid is constructed as a subspace of the composite grid. At the end of the time step, the solution on the composite grid is again restricted to the AMR grid via the constraintΓ.

Multi-resolution lattice Green’s function method

Introduction

Solutions to the Poisson equation are formally solved at each level of the composite mesh using the LGF, before being confined to the AMR grid. We then construct commutative interpolation operators that eliminate the need to explicitly calculate most of the composite grid.

Navier-Stokes LGF solution on a uniform grid

To speed up the evaluation of Eq. 2.6), a variant of the fast multipole method (FMM) is applied. With the IF technique allowing an accurate integration in time of the viscous term, the remaining terms are discretized in time using a semi-ambiguous Runge-Kutta (HERK) method (Hairer, Lubich and Roche, 2006).

Navier-Stokes LGF on an AMR grid

More specifically, given ˆf = ⊗𝑘ˆf𝑘 ∈ RˆQ ⊂ RQ on the AMR grid, the information at level 𝑘 of the composite grid can be estimated by. Together with the constraint operatorΓ one has the approximation relation between the AMR grid and the composite grid.

Figure 2.1: (a) Base unit of a finite-volume staggered grid. (b) 1-D diagram for the composite grid (dotted line) and the corresponding AMR grid (solid line with corresponding vertices in circle)

Adaptivity

The spirit of AMR is to adjust the mesh according to the (measured) smoothness of the solution. To accurately and efficiently estimate the IF convolution we use the following two properties: the IF kernel decays exponentially and the velocity field can be recovered from vorticity. More specifically, to evaluate the IF convolution solution in the region of interestΩsoln.

Implementation

The IF kernel used in AMR, on the other hand, depends not only on the stage of the HERK scheme but also the lattice level𝑘as suggested by Eq. The three-stage HERK scheme considered here requires the application of the AMR-LGF algorithm at each stage, consuming about 60% of the total execution time. The HERK scheme, on the other hand, applies the AMR-IF algorithm to vector fields 5 times, contributing an additional 30% to the total execution time.

Verification

For the numerical tests considered in the following sections, we observe an additional contribution to the execution time of less than 15%. Finally, the reference solution is run using a uniform grid at twice the best test resolution. The computational savings in the spatial adaptability and the nodal distribution will be further discussed in section 2.7.

Collision of vortex rings

Perturbations are added to the initial conditions of each vortex circuit to accelerate the transition. The evolution of the vortex collision taken from the case 𝑁𝑙 = 1 is shown in Fig. showed. As shown in Fig.2.7, the expansion of the vortex ring pair is accompanied by a decay in the kinetic energy and an increase in the enstrophy.

Figure 2.6: Evolution of the vortex ring collision at Re = 7500 from the case 𝑁 𝑙 = 1

Concluding remarks

The combination of the AMR framework and the IB method for external flows with immersed surfaces is introduced in the next chapter. The spherical symmetry of the problem is used in the final step, where the 3D energy spectrum 𝐸e(k) is integrated over a spherical shell to produce a scalar spectrum 𝐸(𝑘). Note, however, that the parallel efficiency is limited to the efficiency of the individual FMMs.

Figure 2.9: Flow visualization and mesh topology at 𝑡 Γ/ 𝑅 2 = 19 . 9 for (a) the non-AMR case ( 𝑁 𝑙 = 0), and (b) the AMR case ( 𝑁 𝑙 = 1)

Immersed boundaries

Introduction

More specifically in the DLM method, the forces associated with points IB are considered as Lagrange multipliers and the no-slip boundary condition is exactly satisfied. 0, 𝑝(x, 𝑡) → 𝑝∞as|x| → ∞, and the unknown forcesf are treated as Lagrange multipliers so that the no-slip constraints are satisfied. In Section 3.2, the original IB-LGF method for a single uniform resolution grid is briefly reviewed.

Immersed boundary method for external flows using LGF on a uni-

The chapter discusses how to discretize and solve the aforementioned equation. 3.1) using a multi-resolution grid and the LGF method. Note that the delta functions in Eq. 3.1b), representing the singular surface forces along the immersed surfaces, are governed by discrete delta functions in R(𝑡) and E(𝑡), where R(𝑡) smears the discrete point IB forces on an offset Cartesian grid, while E (𝑡 ) interpolates the field information from the flux grids back to the IB point. Again, using mimetic properties and commutativity to simplify Eq. where S is related to the Schur complement S𝑖 =EH𝑖.

IB on the AMR grid

Again, we try to approximate the right side of the equation. 3.14) for the composite network using information from the AMR network. Note that the formula IB Eq. 3.14b) and (3.14c), where the former resolves the forces IB and the latter adds an additional contribution to the pressure field. After solving the pressure gradient dGp𝑖 with Eq. 2.14), the updated velocity field at stage 𝑖 after being constrained back to the AMR grid can be expressed as

Adaptivity

Fast linear solver for the IB formulation (Eq. (3.14b))

In the DNS example of flow around a sphere at Re=10,000 to be discussed in the next section, a total number of 3.5×105 IB points are involved in the surface discretization, which makes the application of direct solvers prohibitive. also in parallel. The performance of the iterative CG solver mainly depends on the speed of evaluation of the left operator S𝑖, which involves correlated evaluations of L−C1 and HF via the FLGF method. For the former, we see only a slow increase of the condition number with the number of IB points as reported by Liska and Colonius (2017).

Figure 3.1: Diagram for the static mesh refinement for region near the immersed surface and adaptive mesh refinement with different levels for areas away from the bluff body.

Flow around a sphere

At this Reynolds number, the flow approaches a steady state and thus serves as a verification for the convergence of the velocity. We further demonstrate the algorithm with a DNS of the flow around a sphere at Reynolds numbers Re = 3700 and a preliminary DNS at Re = 10,000. We expect the flow near the surface to be well resolved based on the Re−1/2 scaling of the thickness of the laminar boundary layer.

Figure 3.2: Example of using Fibonacci lattice to distribute IB points on the surface of a sphere

Flow around a delta wing at an angle of attach of 20 ◦

This technique uses a user-defined level setting function of the desired geometry that maps a spatial point to a real number that is negative when the point is inside the geometry, positive when the point is outside the geometry, and zero when the surface. This algorithm first generates a random unstructured mesh for the surface and iteratively improves the mesh toward an equal node distribution by solving a force balance in the element edges and projecting the mesh nodes back to the surface of the geometry using the level set function ( Persson and Strang, 2004 ). The resolution is determined by a low Reynolds number search at Re = 1000 and the scaling of the laminar boundary layer thickness of Re−1/2, which is also similar to the spherical case discussed above.

Figure 3.6: Delta wing setup (He, An, et al., 2019): (a) Experimental setup ; (b) CAD model; (c) immersed boundary mesh.

Concluding remarks

In section 4.4, DNS and LES results are used to visualize the evolution of the turbulence field. In this section we quantify the initial evolution of the turbulence cloud using statistical measurements. Therefore, we used the vorticity formula to calculate the low wavenumber spectra.

Large-eddy simulation

Introduction

In this chapter we model a turbulent flow developing in free space and report on both DNS and LES of this flow. In free space (without submerged boundaries) and without artificial forcing, there are no mechanisms to maintain turbulence, and it will decay over time. In this chapter, we produce an initial condition by first generating isotropic homogeneous turbulence (IHT) in a periodic domain, and initializing a free space.

Stretched vortex sub-grid stress (SGS) model

Due to the spatial adaptability, the total number of calculation cells varies throughout one simulation. Details regarding the efficient evaluation of the aforementioned SGS stress can be found in Voelkl, Pullin and Chan (2000) and Chung and Pullin (2009b). The SVM keeps track of the actual fluid viscosity and also the subgrid kinetic energy, and will automatically become subdominant to actual viscous stress when the flow is locally resolved.

Figure 4.1: Vorticity magnitude in a cross-section through the center at 𝑡 / 𝑡 ℓ = 0 and 𝑡 / 𝑡 ℓ = 8

Problem setup

The spectrum type refers to the leading nonzero order in the low wave limit, which is discussed in detail in Chapter 5. The same resolution (the same number of points used for each length scale 𝐵) is used in the LGF solver for the turbulence cloud. The difference in total kinetic energy is about 0.23% and the maximum relative difference in the spectra for all wavenumbers 𝑘 𝑅 is about 1%.

Qualitative evolution

To study the long-term behavior, we turn to LES calculations of the same setup. To ensure that LES calculations can accurately capture the sustained features of the flow, we qualitatively compare the evolution for DNS and LES of the same case. Nevertheless, over the time span shown in figure 4.3, there is little decorrelation of the large scales in DNS and LES originating from the same initial condition.

Figure 4.2: Energy spectrum of (1) DNS_0 ( ) and (2) a DNS calculation at 3 / 2 times the resolution as that in the case DNS_0 ( ), at 𝑡 / 𝑡 ℓ = 1

Quantitative evolution

This nonuniformity in the radial direction suggests that the energy spectrum as a function of radius 𝑟 and time 𝑡 should be further characterized. Similar to the total energy spectrum 𝐸(𝑘), it requires a relationship between energy and wavenumber, where the wavenumber in a spherical shell is determined using spherical harmonics. The energy decays with time, but the dependence of the shell spectrum on radius is weak.

Figure 4.4: Decay of the kinetic energy E ( 𝑡 ) for different simulations: DNS_0 ( );

Concluding remarks

In this section we investigate the relationship between the size of the vortex rings and properties related to the turbulence. For LES_0, the direction of the initial impulse is indicated by the arrow at the end of the trajectory. For the AMR-IB-LGF solver, we often see that more than 95% of the total computation time is spent on the FFT calculations.

Figure 4.6: Spherical-shell spectrum at different radii and times. The left column (a-c) are results from DNS_0 at 𝑡 / 𝑡 ℓ = 0 , 4

Dynamics and decay of a spherical region of turbulence

Introduction

In this chapter, we further discuss some theoretical points of interest related to the dynamics of the spherical region of turbulence proposed in Chapter 4. In IHT, the development of the largest scales is governed by initial conditions or, in the case of forced IHT, by a forcing scheme. The spherical turbulence region also illustrates the localized turbulence region introduced by Phillips (1956) where the finite viscous evolution rate was theoretically studied.

Initial spectrum and low wavenumber limit

The result of this pulse, as mentioned in Batchelor (1967), is a 1/|x|3 falling shape velocity field. Also plotted is the energy spectrum of the original IHT field scaled by the ratio of the volume of the sphere to the size of the cubic domain𝐵3. We see that the 𝑘−5/3 part of the spectrum from the IHT is preserved in the spherical cloud, while the low-wavenumber behavior is controlled by the resulting pulse (or lack thereof).

Long-term statistics and low wavenumber behavior

Superposition of the energy spectra for the cases LES_0 and LES_IC2 shows that they are similar for 𝑘 𝑅 > 1, which corresponds to a wavelength 𝜆 = 2𝜋 𝑅. The predictability of the long-term evolution from the initial state argues against any universality of the very largest scales of the spherical turbulence cloud. A definition of the center is necessary because, as discussed in § 4.3.1, the final stage of a turbulence cloud is a large vortex ring drifting in the direction of the momentum.

Figure 5.1: Energy spectrum of (1) the initial condition of simulation DNS_0. ( ), where the low wavenumber limit (left of the dotted region) is calculated through an expansion method (Appendix C); (2) DNS_0 with the initial impulse cancelled using Eq

Vortex ring ejections

The first parameter is the width of the transition region associated with the window function,𝜎/𝑅, which is varied from in three cases [LES_D1, LES_0, LES_D2]. We see that the width of the transition region has little influence on the number or scale of the ejections. But, near the edge of the cloud, this imbalance, when directed outward, would push vorticity out of the cloud.

Figure 5.3: Long-term evolution of (a) the kinetic energy decay compared with asymptotic behavior of Saffman IHT E ( 𝑡 ) ∼ 𝑡 − 6 / 5 ( ) and Bathelor IHT E ( 𝑡 ) ∼ 𝑡 − 10 / 7 ( ); (b) the integral scale growth for case LES_0 ( ) and LES_IC2 ( ) up to 𝑡 / 𝑡

Concluding remarks

Advances in hardware architecture also bring new opportunities for future development of the AMR-IB-LGF method. Interestingly, the error vanishes as the ratio of the volume of inhomogeneous turbulence to the volume of homogeneous turbulence. Multiscale geometry and scaling of the turbulent-nonturbulent interface in high-Reynolds-number boundary layers.

Figure 5.8: Long-term turbulence cloud evolution with vortex ring ejections for cases defined in table 4.1

Summary and outlook

Summary

The LGFs are formally applied to all levels of the composite grid with the right-hand side approximated using the information on the AMR grid. To validate the LES, we designed a new turbulent flow in free space - the spherical cloud of turbulence and performed DNS and LES of this flow at Re𝜆 = 122.4. This study indicates that the LES modeling will further improve the AMR-IB-LGF method to simulate flows at higher Reynolds numbers.

Outlook

We showed that the composite grid idea can be easily applied to the immersed boundary method and the resulting IB Navier-Stokes formulation can be efficiently solved using the developed AMR-LGF method, as well as the spatial and refined adaptivity discussed in Chap. 2 Additionally, in Figure A.1b, the dependence of the computation speed with the number of refinement levels is plotted for the case of 𝑁 =6443. As previously mentioned, the complexity scales linearly with the number of levels, and we therefore expect the computational speeds to be independent of the levels of refinement.

Figure A.1: Computational rates and parallel performance.

General definitions

For a function, the spectrum can refer to the magnitude of its Fourier transform and provides information about the scales present in the function. For a random process, on the other hand, the spectrum represents a statistical statement of how the energy is distributed among the scale averages. This form of the spectrum is often written without the expectation operator, but then refers to the spectrum of a deterministic velocity field rather than that of an underlying random process.

Homogeneous and locally homogeneous turbulence

With respect to the transition region, the contribution to the total energy scales with the volume of this region, 4𝜋 𝜎 𝑅2, where 𝜎 is the width of the transition region. An alternative interpretation is that it is linked to the initial/boundary conditions and can be arbitrarily manipulated independently of the turbulence behavior at smaller scales. A more general form in terms of the velocity field for the low wavenumber limit can be derived by expanding equation (C.1). C.5) Comparison of the corresponding terms in equation (C.1) and (C.2) gives a different correlation.