Recent Numerical Advances in Fluid Mechanics

Abstract: A monolithic semi-implicit method is presented for three-dimensional simulation of fluid-structure interaction problems. By using globally continuous finite element for the velocity in the fluid-structure mesh, the continuity of velocity at the.

Problem Statement

Re-engaging the fluid domain improves the quality of the mesh in case of large deformation. The weak formulation of the fluid-structure interaction problem is written in the Eulerian domain, which is unknown, and a fixed-point algorithm solves the global nonlinear problem at each time step.

Updated Lagrangian Framework for the Structure Approximation

We have a relationship between the Cauchy stress tensor of the structureσS and the second Piola–Kirchhoff stress tensorΣ,. The assumption of conservation of mass gives ρS(x,t) =ρJ(X,t)S0(X), where ρS(x,t) is the mass density of the structure in the Euler frame.

Arbitrary Lagrangian Eulerian (ALE) Framework for Approximation of Fluid Equations The Arbitrary Eulerian Lagrangian (ALE) framework is a successful method to solve fluid

Arbitrary Lagrangian Eulerian (ALE) framework for approximating fluid equations Arbitrary Lagrangian Eulerian (ALE) framework is a successful method for solving fluids. We have assumed, for example, that the forcesσF(vF,n+1,pF,n+1)nFon the interface are known.

Monolithic Formulation for the Fluid–Structure Equations

It is possible to use different time discretization schemes, for example Newmark for the structure and implicit Euler for the fluid. Then, the structure including the interface is advanced by the Newmark scheme, and finally the fluid mesh velocity is calculated using the new interface position and velocity.

Numerical Experiments

Tn( x) =x+ (Δt)ϑn+1( x)χΩFn( x) + (Δt) vn+1( x)χΩSn( x) whereχΩFnandχΩSnare the characteristic functions of the fluid and structure fields. i) Domain is calculated explicitly while velocity and pressure are calculated implicitly. Fluid domain parameters (left); and a global mesh for the fluid structure domain (right).

Table 1. The number of vertices, tetrahedra and degrees of freedom (DOF) of fluid–structure linear system for each mesh.

Conclusions

A numerical procedure for fluid-structure interaction with structure displacements constrained by a rigid obstacle.Appl. A stable time-advancement scheme for solving the fluid-structure interaction problem at small structural displacements. Computer science.

On the Kutta Condition in Compressible Flow over Isolated Airfoils

Introduction

It is worth emphasizing that the stream function equation considered in this study is fully equivalent to the full potential equation. Here, the stream function equation is solved in a non-conservative form and can be used to obtain accurate results for subsonic (subcritical) inviscid isentropic compressible flow over isolated airfoils.

Governing Equations

Then we can express the magnitude of the stream function on the outer boundary ψM,jin expression of far-field velocity VM,jas (by considering the equal magnitudes but opposite in sign of stream functions at top and bottom points of EandF, that is, ψF=−candψE=c,cis a constant). We should define the stopping criteria for convergence of solution of the stream function equation (Equation (6)) and density equation (Equation (7)).

Figure 1. The physical and the computational domains. (a) Physical domain. (b) Computational domain.

Results

Test case 5: Airfoil surface pressure coefficient distribution using different grid sizes for NACA 2214 airfoil and M∞=0.55,α=2◦. Test case 7: Airfoil surface pressure coefficient distribution for NACA 2240 airfoil and M∞=0.3 at two different angles of attack α=3◦andα=6◦.

Figure 12. The pressure coefficient distribution for NACA0012 using FOILcom (M ∞ = 0.5, α = 3 ◦ ).

Conclusions

Lewis, R.I.Vortex Element Methods for Fluid Dynamic Analysis of Engineering Systems; Cambridge University Press: Cambridge, UK, 1991. Shapiro, A.H. Dinamika in termodinamika toka stisljive tekočine; John Wiley & Sons: New York, NY, ZDA, 1953.

Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method

• Governing Equation and Cubic B-Spline Galerkin Method The NLS equation is
• Stability Analysis
• Numerical Results and Test Problems
• Conclusions

7] also applied a cubic B-spline finite element method for the numerical solution of the Burger's equation. The approximate solution of the NLS equation was investigated using the Galerkin finite element method with a cubic B-spline shape function.

Table 1. Error norms and conservation laws for single soliton: with h = 0.04, q = 2, S = 4, β = 1.

Effect of Overburden Height on Hydraulic Fracturing of Concrete-Lined Pressure Tunnels Excavated in

Materials and Methods

The forces in the concrete lining due to water pressure are transferred to the surrounding rock. The only degree of freedom inside a water pressure tunnel is the pressure in the fluid nodes. In each analysis, the bearing capacity of the tunnel was determined by increasing the internal pressure (pi) to the point where the first crack formed in the surrounding rock.

Figure 2. The geometry of the surrounding rock and the excavation section.

Results and Discussion 1. Changes in Pore Pressure

Based on the mechanical characteristics of the rock and using the Mohr–Coulomb failure criterion, the failure of the elements in the rock environment surrounding the tunnel was investigated. Figure 8 shows (a) the ultimate bearing capacity of the tunnel in a steady state based on the FEM and NC results, (b) the stress distribution in the rock, and (c) the vertical displacements in the rock. In this regard, the failure of elements in the rock around the lining was investigated by considering the overburden height as well as 70 m.

Figure 6. Numerical vs. analytical results for pore pressures in the rock.

Reynolds Stress Perturbation for Epistemic Uncertainty Quantification of RANS Models

Methodology 1. Epistemic Uncertainty

Λnis is a diagonal tensor containing the eigenvalues ​​of the anisotropy tensor in such an order that λ1>λ2>λ3,. In this paper we focus on the perturbation of the turbulent kinetic energy and the eigenvalues ​​Λ. The distortion of the eigenvalues ​​is realized using the barycentric map proposed by Banerjee et al.

Figure 1. Barycentric map. The limiting states are represented in the vertices as one-component X 1c , two-component X 2c and three-component (isotropic) X 3c turbulence

Results and Analysis

Table 2 summarizes the topology and enhancements used in each of the domain blocks. We can also see that a small amount of perturbation does not properly capture the physics of the problem. These results demonstrate the ability of the framework to capture the uncertainty in the simulation results.

Figure 3. Mesh topology in the flow domain.

Summary and Conclusions

Applying the Reynolds stress perturbation results in the appearance of an additional term (i.e., ΔRij) on the right-hand side of equation (2) as To better describe the Reynolds stress perturbation procedure, three different locations are chosen and their positions on the barycentric map are shown in Figure A3 . Next, the discrepancy between the perturbed Reynolds stress tensor and the original one from the base case (i.e., ΔRij) is estimated (here using a special Matlab code) and added to the right-hand side of the momentum equations in OpenFOAM.

Figure A2. Mean velocity field in the x-direction, normalized with respect to the inlet velocity ( ¯ u/u 0 ) obtained from the baseline case

Shock Capturing in Large Eddy Simulations by Adaptive Filtering

Numerical Method

As mentioned above, an essential requirement of the LES approach adopted here is the use of high-resolution numerical schemes. Here, we use a sixth-order extension of the partition proposed by Hixon and Turkel [23]. When there are shocks, different order filters are applied in the vicinity of the shock.

Figure 1 shows the modified wavenumber ˜ k of the 6/2 scheme. The fourth-order schemes 4/2 and 4/4 that Hixon and Turkel [23] studied are also shown

Basic Tests

Oscillations appear in the vicinity of the shock when the wave is incident at a higher angle (Figure 7c), and are carried into the post-shock region. For example, in the 2D expansion of the Shu–Osher problem, with slightly less filtering, the amplification factor approaches that of linear theory. However, our proposal is not to search for a new, optimal set of filter parameters for each problem, but to e.g. using the set given here.

Table 1. Initial condition for 2D Riemann problem—Case 13 by Lax and Liu [27].

Jet LES

Here, rj= d/2 is the average radius of the jet and δθ is the thickness of the momentum of the jet (taken as rj/20). A comparison of the mean transverse velocity with the experiment along the jet axis is shown in Fig. 11a. Although the usual treatment of shock-capturing flows is to reduce the order of the spatial discretization in the vicinity of the shocks, here the discretization remains the same everywhere and was of sixth order.

Figure 10a is a visualization of the jet in terms of the mean pressure on a plane containing the jet axis, showing the expected cell structures

Cross-Correlation of POD Spatial Modes for the Separation of Stochastic Turbulence and

Coherent Structures

Methodology

Figure 6 shows the parameters that can be used to assess the appropriateness of the number of snapshots considered. It should be noted that when using the POD approach, the ensemble mean is first subtracted. An analysis of the stochastic turbulence characteristics of the flow should be performed on the reconstruction using the remaining 24-1536 POD.

Figure 1. Schematic of LU horizontal water flow tunnel. Indicative measurement plane represented with horizontal green line, downstream of test piece.

Analysis of Separated Velocity Fields

Other turbulent features can be identified in incoherent velocity data by considering the spectral energy content. It is clear that most of the content in the wake region is below 150 Hz, suggesting that the tail in Figures 18 and 19 is related to experimental noise, evident when considering the free-stream region. Phase-Averaged Measurements of Turbulence Properties in the Near Momentum of a High-Reynolds-Number Circular Cylinder by 2C-PIV and 3C-PIV.Exp.

Figure 15. Comparison of fluctuating flow fields for an example flow field. (a) Original flow field;

Numerical Modelling of Air Pollutant Dispersion in Complex Urban Areas: Investigation of City Parts

Methodology 1. Turbulence Model

For all configurations, pollutant dispersion is simulated using LES and dynamic Smagorinsky, SGS, along with the species transport model. This study was performed in the BLASIUS wind tunnel of the Meteorological Institute of the University of Hamburg [30]. The inlet limit is set at 1 m upstream of the buildings, with the ABL profiles measured inside the wind tunnel.

Figure 1. Dimensions of the building models, the source building, and the source [30].

Computational Domain and Boundaries Conditions 1. Geometry and Boundaries Conditions

For the carrier phase boundary conditions, a power law profile is used to describe the variation of the inlet wind speed as a function of height given in equation (5) [33]. The Vortex method is based on the Lagrangian form of the 2D eddy current evolution equation and the Biot-Savart law. The mesh quality has a significant impact on the accuracy of the numerical prediction, as well as the stability of the simulation.

Figure 3. Geometry, boundaries conditions and release locations: (a) Geometry and boundaries conditions used for Hanover city, (b) Release locations used for Hanover city, (c) Geometry and boundaries conditions used for Frankfurt city, (d) Release location

Results and Discussion 1. Hanover City Results

Six different locations are selected to investigate the effect of density on pollutant dispersion. This area is created by the separation of the fluid flow from the edges of the buildings. Figure 10 shows the effect of density ratio on the dispersion of CH4 and CO2.

Figure 5. Characteristics of approaching flow: (a) Velocity at z = 3 m, (b) mean streamwise velocity

Synchronized Multiple Drop Impacts into a Deep Pool

Materials and Methods 1. Experimental Setup

Another critical aspect is the way the drop oscillates at the moment of impact [36,37]. The experiments were performed at atmospheric pressure, at a pool temperature of 30±1◦C and a droplet temperature of 27±1◦C. Regarding the numerical setup, simulations were performed using the interFoam solver of the OpenFOAM® open source CFD tool [38].

Table 1. Investigated conditions.

Results and Discussion

The maximum expansion of the crater decreases when the surface tension is higher (meaning that the straining effect dominates with respect to the higher capillary pressure in the droplet). Figure 11 shows a representation of the crater development at different time points - for case B (w=1.4 m/s), simulated with 12.35 Mcell. Figure 12 shows some rendered frames of crater development for three of these simulated cases.

Figure 2. Superposition of the numerical crater contours to the experimental high-speed photos, for the three investigate cases A (w = 1.0 m/s), B (w = 1.4 m/s), C (w = 2.0 m/s), time up to 8 ms

An Explicit Meshless Point Collocation Solver for Incompressible Navier-Stokes Equations

Algorithm Verification

The flow domain is large relative to the dimensions of the cylinder, as shown in Figure 10. We represent the flow domain with a uniform Cartesian nested grid, locally refined in the vicinity of the cylinder (Figure 11a). We calculate the following flow parameters: the pressure coefficient (Cp) on the body surface, the length (L) of the wake behind the body, the separation angle (θs) and the drag coefficient (CD) of the body.

Table 1. Critical time step for various grid resolution and Reynolds (Re) numbers, using 20 nodes in the support domain.

Numerical Results

In addition, to highlight the versatility of the proposed scheme, we examined the flow in the rectangular channel with multiple (seven in total) cylindrical obstacles. For the finite element model, we solve the flow equations using the generated mesh and. In both cases, we represent the flow domain with uniform embedded Cartesian mesh and irregular nodal distribution (Figure 20).