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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods Governing Equations and Solution Procedure

#### 2.1. Stream Function-Vorticity Formulation of Navier-Stokes Equations

#### 2.2. Temporal Discretization Using Explicit Euler Method

- Step 1
- Update vorticity at the interior grid points$${\omega}^{n+1}={\omega}^{n}-\mathsf{\Delta}t\frac{\partial {\psi}^{n}}{\partial y}\frac{\partial {\omega}^{n}}{\partial x}+\mathsf{\Delta}t\frac{\partial {\psi}^{n}}{\partial x}\frac{\partial {\omega}^{n}}{\partial y}+\mathsf{\Delta}t\frac{1}{Re}\left(\frac{{\partial}^{2}{\omega}^{n}}{\partial {x}^{2}}+\frac{{\partial}^{2}{\omega}^{n}}{\partial {y}^{2}}\right)$$
- Step 2
- Compute stream function by solving Poisson type problem$$\frac{{\partial}^{2}{\psi}^{n+1}}{\partial {x}^{2}}+\frac{{\partial}^{2}{\psi}^{n+1}}{\partial {y}^{2}}=-{\omega}^{n+1}$$
- Step 3
- Update velocity ${\mathit{u}}^{\left(n+1\right)}$$${u}^{n+1}=\frac{\partial {\psi}^{n+1}}{\partial y}$$$${v}^{n+1}=-\frac{\partial {\psi}^{n+1}}{\partial x}$$

#### 2.3. Discretization Corrected Particle Strength Exchange (DC PSE) Method

#### 2.3.1. Particle Strength Exchange (PSE) Operators

**β**| is even and positive when |

**β**| is odd, with

**β**a multi-index [34]. Now, let $\mathit{\beta}=\left({\beta}_{1},{\beta}_{2},\dots ,{\beta}_{n}\right)$, where ${\beta}_{i}$, $i=1,2,\dots ,n$ is a non-negative integer. Then the partial differential operator ${D}^{\mathit{\beta}}$ can be expressed as ${D}^{\mathit{\beta}}=\frac{{\partial}^{\left|\beta \right|}}{\partial {x}_{1}^{{\beta}_{1}}\partial {x}_{2}^{{\beta}_{2}}\dots \partial {x}_{n}^{{\beta}_{n}}}$. The challenge is to find a kernel ${\eta}_{\epsilon}^{\mathit{\beta}}$ that leads to good approximations for D

**. To find such kernels for arbitrary derivatives we adopt the idea in Eldredge et al. [37] and start from the Taylor expansion of a function f(**

^{β}**y**) about a point

**x**:

**x**), depending on whether |

**β**| is odd or even, to both sides and convolute the equation with the unknown kernel ${\eta}_{\epsilon}^{\mathit{\beta}}$. Considering Equation (15), this leads to:

**α**-moments

**f(**

^{β}**x**) with order of accuracy r, the following set of conditions is imposed for the moments M

_{a}:

_{ε}(

**x**) = ${D}^{\mathit{\beta}}f\left(\mathit{x}\right)$ is bounded [34]. The procedure to construct a kernel that satisfies the conditions in Equation (16) has been described in [34]. Once the kernel is defined, the operator in Equation (17) can be discretized using a midpoint quadrature over the nodes as

**x**) is the number of all nodes in a neighborhood around

**x**, which can be defined by a cut-off radius r

_{c}, usually chosen such that N(

**x**) coincides to a certain level of accuracy with the kernel support, and V

_{p}is the volume associated with each particle. Given such a discretization, the discretization error $\u03f5$

_{h}(

**x**) = ${Q}^{\mathit{\beta}}$f(

**x**) − ${Q}_{h}^{\mathit{\beta}}$f(

**x**) is also bounded [34].

#### 2.3.2. The Discretization Corrected PSE Operators

**x**

_{p}) in Equation (22) with its Taylor expansion about

**x**. This leads to the following expression for the derivative approximation:

**x**

_{p}, the coefficients are found by solving the linear system of Equation (28) for

**x**=

**x**

_{p}. Given our choice of kernel function, the DC PSE derivative approximation becomes:

**p**(

**x**) = [p

_{1}(

**x**), p

_{2}(

**x**), p

_{l}(

**x**)] and $\mathit{a}$(

**x**) are the vectors of terms in the monomial basis and their coefficients, respectively. By using the DC PSE method, the spatial derivatives ${\mathit{Q}}^{\beta}$ up to second order are given as:

#### 2.4. Vorticity Boundary Conditions

#### 2.5. Poisson Solver

^{2}) iterations to converge (N being the total number of grid points). Multi-grid algorithms have been developed to accelerate convergence and the computational effort has been significantly reduced [40]. For equally-spaced grids, fast Fourier transform (FFT) solvers are the fastest available algorithms for solving Poisson equations [41].

#### 2.6. Critical Time Step

**,**which includes only first order derivatives. The relative weight of the two operators on the structure of matrix $A$ is dictated by discretisation and velocity field values (velocity is related to stream function and vector potential field values); a more refined discretization leads to a higher weight of operator $\mathit{L}$ (diffusion) and lower influence of operator $\mathit{K}$ (convection), while higher Reynolds number and higher velocity field values leads to a higher influence of operator $\mathit{K}$

**.**When the nodal discretization is not adequately refined, the higher weight of the operator $\mathit{K}$ can lead to eigenvalues with a positive real part; in such case explicit time integration is not possible and discretization has to be refined.

## 3. Algorithm Verification

#### 3.1. Lid-Driven Cavity

_{final}= 250. At that time, the flow regime has all the characteristic features of steady state [42]. We use a $1024\times 1024$ grid resolution and a time step for the simulations of $dt=5\times {10}^{-4}$.

^{−4}, which highlights the accuracy of the proposed method for irregular nodal distributions.

#### 3.2. Backward-Facing Step

^{2}for 0 ≤ y ≤ 0.5, which gives a maximum inflow velocity of u

_{max}= 1.5 and average velocity of u

_{avg}= 1. At the outlet, we assume fully developed flow (du/dx = 0, v = 0). The Reynolds number is defined as Re = u

_{avg}H/v

_{f}, with v

_{f}being the kinematic viscosity.

_{total}= 250 (to ensure that the solution will reach steady state), and the time step dt = 10

^{−4}. The simulation terminates when the NRMSE of the time derivative of the stream function and vorticity field values in two successive time steps is less than 10

^{−6}. The time needed to create the grid was 0.023 s, and it takes 0.3 s to update the solution for each time step. We obtained a steady state solution after 100,000 time steps.

_{lower}≈ 6.1 for the lower wall separation zone, L

_{upper}≈ 5.11 for the upper separation zone, where the separation begins at x ≈ 5.19. Our numerical findings show good agreement with other numerical methods for 2D computations [1,46]. In [1], the authors used a finite difference method and predicted separation lengths of L

_{lower}≈ 6.0 and L

_{upper}≈ 5.75, while [46] using the A Fluid Dynamics Analysis Program (FIDAP) code (Fluid dynamics International Inc., Evanston, IL, USA) predicted L

_{lower}≈ 5.8 and the upper L

_{upper}≈ 4.7.

#### 3.3. Unbounded Flow Past a Cylinder

_{c}= 0.5 and is located at the origin O (0,0) of a square domain with dimensions $-10\le x\le 30$ and $-20\le y\le 20$.

**u**

_{inlet}behaves like the potential flow

**u**

_{potential}given as:

- At the inlet, top and bottom walls (B = inlet, top and bottom)$${\mathit{u}}_{B}={{\mathit{u}}_{potential}|}_{\mathit{B}}$$$${\psi}_{B}={\psi}_{potential}{\rceil}_{B}$$
- At the outlet$$\frac{\partial \psi}{\partial x}=0$$$$\frac{\partial \omega}{\partial x}=0$$
- On the cylindrical surface$${\psi}_{cylinder}=0$$$${\mathit{u}}_{cylinder}=0$$

_{p}) on the body surface, the length (L) of the wake behind the body, the separation angle (θ

_{s}), and the drag coefficient (C

_{D}) of the body. The drag and lift coefficient of the body are given by:

_{rec}) of the wake behind the body, the separation angle (θ

_{s}), and the drag coefficient (C

_{D}) for Re = 40.

## 4. Numerical Results

#### 4.1. Vortex Shedding Behind Cylinders

_{f}= 0.001 m

^{2}/s. The Reynolds number is defined as Re = U

_{m}D/v

_{f}, with the mean velocity U

_{m}given as U

_{m}(x,y,t) = 2U(0,H/2,t)/3. The inflow velocity (x- direction component) is set to $U\left(0,y\right)=4{U}_{m}y\left(\frac{H-y}{{H}^{2}}\right)$, and for U

_{m}= 1.5 m/s yields a Reynolds number Re = 100. At the outlet, we consider fully developed flow (du/dx = 0), while at the remaining boundaries we apply no-slip boundary conditions. The total time for the simulation was set to T

_{tot}= 8 s. The critical time step is computed and monitored for each time step (the calculation involves stream function and vorticity values) to ensure that CFL conditions are fulfilled.

^{−3}s. For each time step, it takes 0.38 s to update the solution.

_{d}of the Cartesian grid in the refined part of the domain. Nodes of the coarse grid have spacing h

_{c}= 2h, and nodes located close to the cylinder (with distance less than 0.25 h

_{d}) are deleted since they affect the condition number of the Vandermonde matrix and decrease the accuracy of the numerical results. The flow domain is represented by 321,897 nodes, which ensure a grid independent solution. For the simulations, we set the time step to dt = 10

^{−3}s.

_{tot}= 8 s, the time step was set to dt = 10

^{−4}s. For each time step, the time needed to update the solution was 0.32 s. Figure 17 displays the stream function and vorticity contours for Re = 100 at time t = 1 and t = 2 s.

_{x}, u

_{y}), considering as gold standard the velocity field values calculated using FEniCS. Figure 18 shows the NRMSE for the velocity components for the time interval between t = 0 and t = 1 s.

#### 4.2. Flow in a Duct with Irregular Geometry

^{3}. The inflow condition ${U}_{inlet}=4{U}_{m}y\left(\frac{H-y}{{H}^{2}}\right)$, with U

_{m}= 0.035 m/s. The Reynolds number $Re=\rho {U}_{m}H/\mu $, with D being a characteristic length defined separately for each case. At the outlet, the flow is fully developed (du/dx = 0). For the remaining boundaries we applied no-slip conditions. In both cases, we represent the flow domain with uniform Cartesian embedded grid, and irregular nodal distribution (Figure 20).

_{tot}= 30 s and the time step used was dt = 10

^{–4}s (for each time step the solution is computed in 0.35 s). Figure 21 shows the iso-contours for the stream function and vorticity field functions, at different time instances: t = 0.5 and t = 1 s, and for steady state (steady state is reached when, for two successive time instances ${t}^{n+1}$ and ${t}^{n}$ for both vorticity and stream function field values, $NRMSE/dt<{10}^{-8}$).

_{tot}= 30 s and the time step used is dt = 10

^{−4}s, which is smaller than the critical time that ensures stability for the explicit solver (the solution in each time step is computed in 0.52 s). The inflow velocity ${U}_{inlet}=4{U}_{m}y\left(\frac{H-y}{{H}^{2}}\right)$, with U

_{m}= 0.015 m/s results in Reynolds number Re = 212. Figure 23 shows the iso-contours for the stream function and vorticity field functions, at time t = 0.5 and t = 1 s and for steady state (steady state is reached when, for two successive time instances ${t}^{n+1}$ and ${t}^{n}$ for both vorticity and stream function field values, $NRMSE/dt<{10}^{-8}$).

_{x}and u

_{y}) on the vertices of the triangular elements (the same nodes are used in the meshless point collocation flow solver). Figure 24 shows the NRMSE for the velocity components (bypass and bifurcation flow examples) for the time interval between t = 0 and t = 1 s.

## 5. Conclusions

- Rapid and easy generation of computational grids, as demonstrated by examples of the flow in vortex shedding behind cylinders and flow in a duct with irregular geometry in Section 4.1 and Section 4.2, respectively.
- High accuracy, as demonstrated in verification examples discussed in Section 3.
- Computational efficiency, as demonstrated by time per iteration (Table 3) and total simulation time given for all examples in this paper. Our approach makes Direct Numerical Simulations (DNS) [54] possible (see lid driven cavity example with $1024\times 1024$ grid resolution), even on personal computers.
- Critical time step is easily computed with low computational cost.
- The imposition of Dirichlet boundary conditions is straightforward.
- Easy and straightforward way to impose vorticity boundary conditions using spatial derivatives computed using the DC PSE method (Equation (33))
- Simplicity: we use a MATLAB code of ca. 150 lines to solve flow equations, and a C++ code of ca. 140 lines to compute spatial derivatives.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Velocity profiles along the vertical line passing through the geometric center of the cavity for u- velocity component (left) and horizontal line passing through the geometric center of the cavity for the v- velocity component (right) for (

**a**) Re = 7500 and (

**b**) Re = 10,000. The results we obtained in this study using our methods described in section Methods Governing Equations and Solution Procedure (indicated as “Meshless” in the velocity profiles) are compared with the results reported in Ghia et al. [42].

**Figure 3.**Streamline contours of primary, secondary and additional corner vortices for lid-driven cavity flow with Re = 7500.

**Figure 4.**Streamline contours of primary, secondary and additional corner vortices for lid-driven cavity flow with Re = 10,000.

**Figure 7.**Contour plots of the (

**a**) stream function and (

**b**) vorticity for Re = 800 for the backward-facing step flow problem using the stream function and vorticity values reported in Gartling [45].

**Figure 8.**(

**a**) Horizontal and (

**b**) vertical velocity profiles, and (

**c**) vorticity profiles at x = 7 and x = 15 for the backward-facing step flow problem with Re = 800.

**Figure 9.**(

**a**) Shear stress for the upper and lower wall for Re = 800 for the backward-facing step flow problem (

**b**) convergence of the upper wall shear stress with successively denser point clouds (40, 80 and 120 correspond to the number of nodes in the y-direction).

**Figure 11.**(

**a**) Locally refined Cartesian and (

**b**) irregular nodal configurations for the flow past cylinder flow problem.

**Figure 12.**(

**a**) Stream function and (

**b**) vorticity contours around the body for flow behind a cylinder using a Cartesian embedded nodal distribution.

**Figure 14.**(

**a**) Stream function and (

**b**) vorticity contours for flow around a cylinder at time t = 2, 4 and 8 s.

**Figure 16.**(

**a**) Cartesian embedded and (

**b**) irregular nodal distributions used in duct flow problems considering multiple cylindrical obstacles.

**Figure 17.**(

**a)**Stream function and (

**b**) vorticity contours for flow behind multiple cylinders at time t = 1 and t = 2 s.

**Figure 18.**Normalized Root Mean Square Error (NRMSE) for the velocity components between the proposed scheme and the finite element solution.

**Figure 20.**(

**a**) Cartesian embedded and (

**b**) irregular nodal distribution for the complex flow geometries.

**Figure 21.**Stream function and vorticity contours for the bypass test case at (

**a**) t = 0.5 s and (

**b**) t = 1 s and (

**c**) steady state.

**Figure 22.**(

**a**) u-velocity profiles at x = 0.07 at time t = 3 s for the coarse, moderate and dense nodal distributions for Re = 474 (U

_{m}= 0.035 m/s) and (

**b**) u-velocity profiles at x = 0.07 and 0.08 for flow in the bypass geometry.

**Figure 24.**Normalized Root Mean Square Error (NRMSE) for the velocity components between the proposed scheme and the finite element solution for (

**a**) bypass and (

**b**) bifurcation flow cases.

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

Grid Resolution | Re = 5000 | Re = 7500 | Re = 10,000 |
---|---|---|---|

$601\times 601$ | $3.4\times {10}^{-3}$ | $5.1\times {10}^{-3}$ | $6.7\times {10}^{-3}$ |

$1024\times 1024$ | $1.1\times {10}^{-3}$ | $1.6\times {10}^{-3}$ | $2\times {10}^{-3}$ |

$2048\times 2048$ | $3.1\times {10}^{-4}$ | $4.4\times {10}^{-4}$ | $5\times {10}^{-4}$ |

**Table 2.**Critical time step for various grid resolution and Re numbers, using 40 nodes in the support domain.

Grid Resolution | Re = 5000 | Re = 7500 | Re = 10,000 |
---|---|---|---|

$601\times 601$ | $7.6\times {10}^{-3}$ | $8.2\times {10}^{-3}$ | $9.2\times {10}^{-3}$ |

$1024\times 1024$ | $3.1\times {10}^{-3}$ | $3.5\times {10}^{-3}$ | $3.7\times {10}^{-3}$ |

$2048\times 2048$ | $1.2\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $9.1\times {10}^{-4}$ |

**Table 3.**Computational time (in seconds) for computing (

**1**) spatial derivatives, (

**2**) factorization and (

**3**) numerical solution for various grid resolutions.

Grid Resolution | 1. Derivatives | 2. Factorization | 3. Solution (in s)/Iteration |
---|---|---|---|

$601\times 601$ | 3.92 s | 4.97 s | 0.22 s |

$1024\times 1024$ | 11.10 s | 21.35 s | 0.67 s |

$2048\times 2048$ | 56.79 s | 274.70 s | 41.39 s |

Stream Function | Vorticity | ||||
---|---|---|---|---|---|

Contour Level | Value of ψ | Contour Number | Value of ψ | Contour Number | Value of ψ |

a | −1.0 $\times $ 10^{−10} | ||||

b | −1.0 $\times $ 10^{−7} | 0 | 1.0 $\times $ 10^{−8} | ||

c | −1.0 $\times $ 10^{−5} | 1 | 1.0 $\times $ 10^{−7} | ||

d | −1.0 $\times $ 10^{−4} | 2 | 1.0 $\times $ 10^{−6} | 0 | 0 |

e | −0.0100 | 3 | 1.0 $\times $ 10^{−5} | $\pm $1 | $\pm $0.5 |

f | −0.0300 | 4 | 5.0 $\times $ 10^{−5} | $\pm $2 | $\pm $1.0 |

g | −0.0500 | 5 | 1.0 $\times $ 10^{−4} | $\pm $3 | $\pm $2.0 |

h | −0.0700 | 6 | 2.5 $\times $ 10^{−4} | $\pm $4 | $\pm $3.0 |

i | −0.0900 | 7 | 5.0 $\times $ 10^{−4} | 5 | 4.0 |

j | −0.1000 | 8 | 1.0 $\times $ 10^{−3} | 6 | 5.0 |

k | −0.1100 | 9 | 1.5 $\times $ 10^{−3} | ||

l | −0.1150 | 10 | 3.0 $\times $ 10^{−3} | ||

m | −0.1175 |

**Table 5.**Comparison of center of primary, secondary, and ternary vortices, and the corresponding stream function values.

Re | Reference | ${\mathbf{\Psi}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$ | (x_{1st}, y_{1st}) | ${\mathbf{\Psi}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ | (x_{2nd}, y_{2nd}) | ${\mathbf{\Psi}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$ | (x_{3rd}, y_{3rd}) | ${\mathbf{\Psi}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ | (x_{4th}, y_{4th}) |
---|---|---|---|---|---|---|---|---|---|

7500 | Ghia [42] | −0.1199 | (0.5117, 0.5322) | $3.30\times {10}^{-3}$ | (0.7813, 0.0625) | $1.47\times {10}^{-3}$ | (0.0645, 0.1504) | $2.05\times {10}^{-3}$ | (0.0664, 0.9141) |

Present | −0.1132 | (0.5100,0.5333) | $3.12\times {10}^{-3}$ | (0.7783, 0.0634) | $1.36\times {10}^{-3}$ | (0.0612, 0.1554) | $1.97\times {10}^{-3}$ | (0.0614, 0.9267) | |

10,000 | Ghia [42] | −0.1197 | (0.5117, 0.5333) | $3.42\times {10}^{-3}$ | (0.7656, 0.0586) | $1.52\times {10}^{-3}$ | (0.0586, 0.1641) | $2.42\times {10}^{-3}$ | (0.0703, 0.9141) |

Present | −0.1183 | (0.5109, 0.5331) | $3.36\times {10}^{-3}$ | (0.7642, 0.0582) | $1.51\times {10}^{-3}$ | (0.0583, 0.1643) | $2.41\times {10}^{-3}$ | (0.0703, 0.9142) |

Re = 7500 | Re = 10,000 | |||
---|---|---|---|---|

y | Ghia [42] | Present | Ghia [42] | Present |

1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 |

0.9766 | 0.47244 | 0.46956 | 0.47221 | 0.47244 |

0.9688 | 0.47048 | 0.46818 | 0.47783 | 0.47797 |

0.9609 | 0.47323 | 0.47186 | 0.48070 | 0.48034 |

0.9531 | 0.47167 | 0.47184 | 0.47804 | 0.47814 |

0.8516 | 0.34228 | 0.3381 | 0.34635 | 0.34685 |

0.7344 | 0.20591 | 0.19527 | 0.20673 | 0.20665 |

0.6172 | 0.08342 | 0.08142 | 0.08344 | 0.08332 |

0.5000 | −0.03800 | −0.03798 | 0.03111 | 0.03154 |

0.4531 | −0.07503 | −0.07245 | −0.07540 | −0.07534 |

0.2813 | −0.23176 | −0.23905 | −0.23186 | −0.23136 |

0.1719 | −0.32393 | −0.31801 | −0.32709 | −0.32729 |

0.1016 | −0.38324 | −0.38291 | −0.38000 | −0.38012 |

0.0703 | −0.43025 | −0.42988 | −0.41657 | −0.41648 |

0.0625 | −0.4359 | −0.43050 | −0.42537 | −0.42587 |

0.0547 | −0.43154 | −0.42935 | −0.42735 | −0.42775 |

0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Re = 7500 | Re = 10,000 | |||
---|---|---|---|---|

x | Ghia [42] | Present | Ghia [42] | Present |

1.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

0.9688 | −0.53858 | −0.53889 | −0.54302 | −0.54367 |

0.9609 | −0.55216 | −0.55258 | −0.52487 | −0.52432 |

0.9531 | −0.52347 | −0.52363 | −0.49099 | −0.49056 |

0.9453 | −0.4859 | −0.48576 | −0.45863 | −0.45834 |

0.9063 | −0.41050 | −0.41082 | −0.41496 | −0.41439 |

0.8594 | −0.36213 | −0.36257 | −0.36737 | −0.36756 |

0.8047 | −0.30448 | −0.30432 | −0.30719 | −0.30739 |

0.5000 | 0.00824 | 0.00851 | 0.00831 | 0.00845 |

0.2344 | 0.27348 | 0.27378 | 0.27224 | 0.27267 |

0.2266 | 0.28117 | 0.28135 | 0.28003 | 0.28056 |

0.1563 | 0.3506 | 0.35082 | 0.35070 | 0.35078 |

0.0938 | 0.41824 | 0.41867 | 0.41487 | 0.41463 |

0.0781 | 0.43564 | 0.43591 | 0.43124 | 0.43165 |

0.0703 | 0.4403 | 0.44059 | 0.43733 | 0.43745 |

0.0625 | 0.43979 | 0.43962 | 0.43983 | 0.43953 |

0.0000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

y | u | $\mathit{v}\text{}(\times {10}^{-2})$ | $\mathit{\omega}$ | |||
---|---|---|---|---|---|---|

Gartling [45] | Present | Gartling [45] | Present | Gartling [45] | Present | |

0.50 | 0.000 | 0.000 | 0.000 | 0.000 | −1.034 | −1.033 |

0.45 | −0.038 | −0.038 | −0.027 | −0.027 | −0.493 | −0.491 |

0.40 | −0.049 | −0.049 | −0.086 | −0.085 | 0.061 | 0.065 |

0.35 | −0.032 | −0.031 | −0.147 | −0.144 | 0.635 | 0.641 |

0.30 | 0.015 | 0.016 | −0.193 | −0.186 | 1.237 | 1.245 |

0.25 | 0.092 | 0.093 | −0.225 | −0.213 | 1.888 | 1.894 |

0.20 | 0.204 | 0.205 | −0.268 | 0.249 | 2.588 | 2.585 |

0.15 | 0.349 | 0.349 | −0.362 | −0.333 | 3.267 | 3.246 |

0.10 | 0.522 | 0.523 | −0.544 | −0.502 | 3.751 | 3.709 |

0.05 | 0.709 | 0.709 | −0.823 | −0.767 | 3.821 | 3.767 |

0.00 | 0.885 | 0.885 | −1.165 | −1.095 | 3.345 | 3.295 |

−0.05 | 1.024 | 1.024 | −1.507 | −1.425 | 2.362 | 2.328 |

−0.10 | 1.105 | 1.104 | −1.778 | −1.691 | 1.046 | 1.034 |

−0.15 | 1.118 | 1.117 | −1.925 | −1.838 | −0.374 | −0.367 |

−0.20 | 1.062 | 1.061 | −1.917 | −1.836 | −1.684 | −1.661 |

−0.25 | 0.948 | 0.947 | −1.748 | −1.678 | −2.719 | −2.687 |

−0.30 | 0.792 | 0.792 | −1.423 | −1.378 | −3.392 | −3.355 |

−0.35 | 0.613 | 0.613 | −1.000 | −0.960 | −3.658 | −3.636 |

−0.4 | 0.428 | 0.427 | −0.504 | −0.482 | −3.687 | −3.719 |

−0.45 | 0.232 | 0.232 | −0.118 | −0.113 | −4.132 | −4.177 |

−0.50 | 0.000 | 0.000 | 0.000 | 0.000 | −5.140 | −5.146 |

y | u | $\mathit{v}\text{}(\times {10}^{-2})$ | $\mathit{\omega}$ | |||
---|---|---|---|---|---|---|

Gartling [45] | Present | Gartling [45] | Present | Gartling [45] | Present | |

0.50 | 0.000 | 0.000 | 0.000 | 0.000 | 2.027 | 2.033 |

0.45 | 0.101 | 0.101 | 0.021 | 0.021 | 2.013 | 2.018 |

0.40 | 0.202 | 0.202 | 0.072 | 0.072 | 2.023 | 2.027 |

0.35 | 0.304 | 0.304 | 0.140 | 0.139 | 2.058 | 2.058 |

0.30 | 0.408 | 0.408 | 0.207 | 0.206 | 2.090 | 2.085 |

0.25 | 0.512 | 0.512 | 0.260 | 0.259 | 2.075 | 2.063 |

0.20 | 0.613 | 0.613 | 0.288 | 0.286 | 1.959 | 1.943 |

0.15 | 0.704 | 0.704 | 0.283 | 0.281 | 1.703 | 1.687 |

0.10 | 0.779 | 0.779 | 0.245 | 0.243 | 1.298 | 1.283 |

0.05 | 0.831 | 0.831 | 0.180 | 0.177 | 0.761 | 0.751 |

0.00 | 0.853 | 0.853 | 0.095 | 0.093 | 0.141 | 0.137 |

−0.05 | 0.844 | 0.844 | 0.003 | 0.002 | −0.500 | −0.447 |

−0.10 | 0.804 | 0.804 | −0.081 | −0.081 | −1.096 | −1.086 |

−0.15 | 0.737 | 0.737 | −0.147 | −0.147 | −1.588 | −1.574 |

−0.20 | 0.649 | 0.649 | −0.185 | −0.185 | −1.939 | −1.923 |

−0.25 | 0.547 | 0.547 | −0.191 | −0.191 | −2.139 | −2.125 |

−0.30 | 0.438 | 0.438 | −0.166 | −0.166 | −2.213 | −2.203 |

−0.35 | 0.328 | 0.328 | −0.119 | −0.119 | −2.210 | −2.206 |

−0.40 | 0.218 | 0.218 | −0.065 | −0.065 | −2.184 | −2.185 |

−0.45 | 0.109 | 0.109 | −0.019 | −0.019 | −2.174 | 2.177 |

−0.50 | 0.000 | 0.000 | 0.000 | 0.000 | −2.185 | −2.189 |

**Table 10.**Comparison of the wake length (L

_{sep}), the separation angle (θ

_{sep}), and the drag coefficient (C

_{D}) for Reynolds number Re = 40.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bourantas, G.C.; Zwick, B.F.; Joldes, G.R.; Loukopoulos, V.C.; Tavner, A.C.R.; Wittek, A.; Miller, K.
An Explicit Meshless Point Collocation Solver for Incompressible Navier-Stokes Equations. *Fluids* **2019**, *4*, 164.
https://doi.org/10.3390/fluids4030164

**AMA Style**

Bourantas GC, Zwick BF, Joldes GR, Loukopoulos VC, Tavner ACR, Wittek A, Miller K.
An Explicit Meshless Point Collocation Solver for Incompressible Navier-Stokes Equations. *Fluids*. 2019; 4(3):164.
https://doi.org/10.3390/fluids4030164

**Chicago/Turabian Style**

Bourantas, George C., Benjamin F. Zwick, Grand R. Joldes, Vassilios C. Loukopoulos, Angus C. R. Tavner, Adam Wittek, and Karol Miller.
2019. "An Explicit Meshless Point Collocation Solver for Incompressible Navier-Stokes Equations" *Fluids* 4, no. 3: 164.
https://doi.org/10.3390/fluids4030164