+++ date = 2021-07-10T08:23:47Z description = "" draft = false slug = "jeonsanyuceyeoghag-cfd-with-python-navier-stokes-equation" title = "[전산유체역학] CFD with Python (Navier-Stokes Equation)" +++ ## 1-D Linear Convection 1차원 선형 열전도 방정식은 가장 심플하면서도 가장 기초적인 방정식입니다. $$ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0 $$ 이 식을 오일러 방정식으로 변환하여 수치해석적으로 해를 구할 수 있도록 변환을 해줍니다. $$ u_i^{n+1} = u_i^n - c \frac{\Delta t}{\Delta x}(u_i^n-u_{i-1}^n) $$ 이제 이 오일러 방정식을 파이썬으로 구현해봅니다. ```python import numpy from matplotlib import pyplot import time, sys %matplotlib inline nx = 41 # try changing this number from 41 to 81 and Run All ... what happens? dx = 2 / (nx-1) nt = 25 #nt is the number of timesteps we want to calculate dt = .025 #dt is the amount of time each timestep covers (delta t) c = 1 #assume wavespeed of c = 1 u = numpy.ones(nx) #numpy function ones() u[int(.5 / dx):int(1 / dx + 1)] = 2 #setting u = 2 between 0.5 and 1 as per our I.C.s un = numpy.ones(nx) #initialize a temporary array for n in range(nt): #loop for values of n from 0 to nt, so it will run nt times un = u.copy() ##copy the existing values of u into un for i in range(1, nx): ## you can try commenting this line and... #for i in range(nx): ## ... uncommenting this line and see what happens! u[i] = un[i] - c * dt / dx * (un[i] - un[i-1]) pyplot.plot(numpy.linspace(0, 2, nx), u); ``` {{< figure src="https://blog.kakaocdn.net/dn/ZbM8j/btq9fWovXzY/D1HOqkCqgkw9YLDpyMFxb1/img.png" >}} ## 1-D Convection Equation (Non-Linear) $$ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0 $$ $$ u_i^{n+1} = u_i^n - u_i^n \frac{\Delta t}{\Delta x} (u_i^n - u_{i-1}^n) $$ ```python import numpy # we're importing numpy from matplotlib import pyplot # and our 2D plotting library %matplotlib inline nx = 41 dx = 2 / (nx - 1) nt = 20 #nt is the number of timesteps we want to calculate dt = .025 #dt is the amount of time each timestep covers (delta t) u = numpy.ones(nx) #as before, we initialize u with every value equal to 1. u[int(.5 / dx) : int(1 / dx + 1)] = 2 #then set u = 2 between 0.5 and 1 as per our I.C.s un = numpy.ones(nx) #initialize our placeholder array un, to hold the time-stepped solution for n in range(nt): #iterate through time un = u.copy() ##copy the existing values of u into un for i in range(1, nx): ##now we'll iterate through the u array ###This is the line from Step 1, copied exactly. Edit it for our new equation. ###then uncomment it and run the cell to evaluate Step 2 ###u[i] = un[i] - c * dt / dx * (un[i] - un[i-1]) pyplot.plot(numpy.linspace(0, 2, nx), u) ##Plot the results ``` ## 1-D Diffusion Equation $$ \frac{\partial u}{\partial t}= \nu \frac{\partial^2 u}{\partial x^2} $$ $$ u_{i}^{n+1}=u_{i}^{n}+\frac{\nu\Delta t}{\Delta x^2}(u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}) $$ ```python import numpy #loading our favorite library from matplotlib import pyplot #and the useful plotting library %matplotlib inline nx = 41 dx = 2 / (nx - 1) nt = 20 #the number of timesteps we want to calculate nu = 0.3 #the value of viscosity sigma = .2 #sigma is a parameter, we'll learn more about it later dt = sigma * dx**2 / nu #dt is defined using sigma ... more later! u = numpy.ones(nx) #a numpy array with nx elements all equal to 1. u[int(.5 / dx):int(1 / dx + 1)] = 2 #setting u = 2 between 0.5 and 1 as per our I.C.s un = numpy.ones(nx) #our placeholder array, un, to advance the solution in time for n in range(nt): #iterate through time un = u.copy() ##copy the existing values of u into un for i in range(1, nx - 1): u[i] = un[i] + nu * dt / dx**2 * (un[i+1] - 2 * un[i] + un[i-1]) pyplot.plot(numpy.linspace(0, 2, nx), u); ``` {{< figure src="https://blog.kakaocdn.net/dn/oaKgN/btq9iA57vBH/weCFyYoImjFkasFiDMir3k/img.png" >}} ## Burger's Equation $$ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial ^2u}{\partial x^2} $$ $$ u_i^{n+1} = u_i^n - u_i^n \frac{\Delta t}{\Delta x} (u_i^n - u_{i-1}^n) + \nu \frac{\Delta t}{\Delta x^2}(u_{i+1}^n - 2u_i^n + u_{i-1}^n) $$ ```python import numpy import sympy from sympy import init_printing from matplotlib import pyplot from sympy.utilities.lambdify import lambdify %matplotlib inline init_printing(use_latex=True) x, nu, t = sympy.symbols('x nu t') phi = (sympy.exp(-(x - 4 * t)**2 / (4 * nu * (t + 1))) + sympy.exp(-(x - 4 * t - 2 * sympy.pi)**2 / (4 * nu * (t + 1)))) phiprime = phi.diff(x) u = -2 * nu * (phiprime / phi) + 4 ufunc = lambdify((t, x, nu), u) ###variable declarations nx = 101 nt = 100 dx = 2 * numpy.pi / (nx - 1) nu = .07 dt = dx * nu x = numpy.linspace(0, 2 * numpy.pi, nx) un = numpy.empty(nx) t = 0 u = numpy.asarray([ufunc(t, x0, nu) for x0 in x]) for n in range(nt): un = u.copy() for i in range(1, nx-1): u[i] = un[i] - un[i] * dt / dx *(un[i] - un[i-1]) + nu * dt / dx**2 *\ (un[i+1] - 2 * un[i] + un[i-1]) u[0] = un[0] - un[0] * dt / dx * (un[0] - un[-2]) + nu * dt / dx**2 *\ (un[1] - 2 * un[0] + un[-2]) u[-1] = u[0] u_analytical = numpy.asarray([ufunc(nt * dt, xi, nu) for xi in x]) pyplot.figure(figsize=(11, 7), dpi=100) pyplot.plot(x,u, marker='o', lw=2, label='Computational') pyplot.plot(x, u_analytical, label='Analytical') pyplot.xlim([0, 2 * numpy.pi]) pyplot.ylim([0, 10]) pyplot.legend(); ``` {{< figure src="https://blog.kakaocdn.net/dn/cE7P8B/btq9dF2BVpt/J6GbNhRT4dX1nfB2GPkurK/img.png" >}} ## 2-D Linear Convection $$ \frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x} + c\frac{\partial u}{\partial y} = 0 $$ $$ u_{i,j}^{n+1} = u_{i,j}^n-c \frac{\Delta t}{\Delta x}(u_{i,j}^n-u_{i-1,j}^n)-c \frac{\Delta t}{\Delta y}(u_{i,j}^n-u_{i,j-1}^n) $$ ```python from mpl_toolkits.mplot3d import Axes3D ##New Library required for projected 3d plots import numpy from matplotlib import pyplot, cm %matplotlib inline ###variable declarations nx = 81 ny = 81 nt = 100 c = 1 dx = 2 / (nx - 1) dy = 2 / (ny - 1) sigma = .2 dt = sigma * dx x = numpy.linspace(0, 2, nx) y = numpy.linspace(0, 2, ny) u = numpy.ones((ny, nx)) ##create a 1xn vector of 1's un = numpy.ones((ny, nx)) ## ###Assign initial conditions ##set hat function I.C. : u(.5<=x<=1 && .5<=y<=1 ) is 2 u[int(.5 / dy):int(1 / dy + 1),int(.5 / dx):int(1 / dx + 1)] = 2 ###Plot Initial Condition ##the figsize parameter can be used to produce different sized images fig = pyplot.figure(figsize=(11, 7), dpi=100) ax = fig.gca(projection='3d') X, Y = numpy.meshgrid(x, y) surf = ax.plot_surface(X, Y, u[:], cmap=cm.viridis) ``` {{< figure src="https://blog.kakaocdn.net/dn/MH3sO/btq9fmIfvXs/Tbral2sgJxUQHfJgs1hG61/img.png" >}} ## 2-D Convection $$ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = 0 $$$$ \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = 0 $$$$ u_{i,j}^{n+1} = u_{i,j}^n - u_{i,j} \frac{\Delta t}{\Delta x} (u_{i,j}^n-u_{i-1,j}^n) - v_{i,j}^n \frac{\Delta t}{\Delta y} (u_{i,j}^n-u_{i,j-1}^n) $$$$ v_{i,j}^{n+1} = v_{i,j}^n - u_{i,j} \frac{\Delta t}{\Delta x} (v_{i,j}^n-v_{i-1,j}^n) - v_{i,j}^n \frac{\Delta t}{\Delta y} (v_{i,j}^n-v_{i,j-1}^n) $$ ```python from mpl_toolkits.mplot3d import Axes3D from matplotlib import pyplot, cm import numpy %matplotlib inline ###variable declarations nx = 101 ny = 101 nt = 80 c = 1 dx = 2 / (nx - 1) dy = 2 / (ny - 1) sigma = .2 dt = sigma * dx x = numpy.linspace(0, 2, nx) y = numpy.linspace(0, 2, ny) u = numpy.ones((ny, nx)) ##create a 1xn vector of 1's v = numpy.ones((ny, nx)) un = numpy.ones((ny, nx)) vn = numpy.ones((ny, nx)) ###Assign initial conditions ##set hat function I.C. : u(.5<=x<=1 && .5<=y<=1 ) is 2 u[int(.5 / dy):int(1 / dy + 1), int(.5 / dx):int(1 / dx + 1)] = 2 ##set hat function I.C. : v(.5<=x<=1 && .5<=y<=1 ) is 2 v[int(.5 / dy):int(1 / dy + 1), int(.5 / dx):int(1 / dx + 1)] = 2 fig = pyplot.figure(figsize=(11, 7), dpi=100) ax = fig.gca(projection='3d') X, Y = numpy.meshgrid(x, y) ax.plot_surface(X, Y, u, cmap=cm.viridis, rstride=2, cstride=2) ax.set_xlabel('$x$') ax.set_ylabel('$y$'); ``` {{< figure src="https://blog.kakaocdn.net/dn/bbiKmO/btq9hyAHR99/3KeBvxXPvCzXXYqrTlSj9k/img.png" >}} ## 2D Diffusion $$ \frac{\partial u}{\partial t} = \nu \frac{\partial ^2 u}{\partial x^2} + \nu \frac{\partial ^2 u}{\partial y^2} $$$$ \begin{split}u_{i,j}^{n+1} = u_{i,j}^n &+ \frac{\nu \Delta t}{\Delta x^2}(u_{i+1,j}^n - 2 u_{i,j}^n + u_{i-1,j}^n) \\&+ \frac{\nu \Delta t}{\Delta y^2}(u_{i,j+1}^n-2 u_{i,j}^n + u_{i,j-1}^n)\end{split} $$ ```python import numpy from matplotlib import pyplot, cm from mpl_toolkits.mplot3d import Axes3D ##library for 3d projection plots %matplotlib inline ###variable declarations nx = 31 ny = 31 nt = 17 nu = .05 dx = 2 / (nx - 1) dy = 2 / (ny - 1) sigma = .25 dt = sigma * dx * dy / nu x = numpy.linspace(0, 2, nx) y = numpy.linspace(0, 2, ny) u = numpy.ones((ny, nx)) # create a 1xn vector of 1's un = numpy.ones((ny, nx)) ###Assign initial conditions # set hat function I.C. : u(.5<=x<=1 && .5<=y<=1 ) is 2 u[int(.5 / dy):int(1 / dy + 1),int(.5 / dx):int(1 / dx + 1)] = 2 ###Run through nt timesteps def diffuse(nt): u[int(.5 / dy):int(1 / dy + 1),int(.5 / dx):int(1 / dx + 1)] = 2 for n in range(nt + 1): un = u.copy() u[1:-1, 1:-1] = (un[1:-1,1:-1] + nu * dt / dx**2 * (un[1:-1, 2:] - 2 * un[1:-1, 1:-1] + un[1:-1, 0:-2]) + nu * dt / dy**2 * (un[2:,1: -1] - 2 * un[1:-1, 1:-1] + un[0:-2, 1:-1])) u[0, :] = 1 u[-1, :] = 1 u[:, 0] = 1 u[:, -1] = 1 fig = pyplot.figure() ax = fig.gca(projection='3d') surf = ax.plot_surface(X, Y, u[:], rstride=1, cstride=1, cmap=cm.viridis, linewidth=0, antialiased=True) ax.set_zlim(1, 2.5) ax.set_xlabel('$x$') ax.set_ylabel('$y$'); diffuse(14) ``` {{< figure src="https://blog.kakaocdn.net/dn/eLwQEW/btq9e0ysgnx/YVHruuNPlMa6pJODaIGJdK/img.png" >}} ## Burgers' Equation in 2D $$ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \; \left(\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2}\right) $$$$ \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = \nu \; \left(\frac{\partial ^2 v}{\partial x^2} + \frac{\partial ^2 v}{\partial y^2}\right) $$$$ \begin{split}v_{i,j}^{n+1} = & v_{i,j}^n - \frac{\Delta t}{\Delta x} u_{i,j}^n (v_{i,j}^n - v_{i-1,j}^n) - \frac{\Delta t}{\Delta y} v_{i,j}^n (v_{i,j}^n - v_{i,j-1}^n) \\&+ \frac{\nu \Delta t}{\Delta x^2}(v_{i+1,j}^n-2v_{i,j}^n+v_{i-1,j}^n) + \frac{\nu \Delta t}{\Delta y^2} (v_{i,j+1}^n - 2v_{i,j}^n + v_{i,j-1}^n)\end{split} $$ ```python import numpy from matplotlib import pyplot, cm from mpl_toolkits.mplot3d import Axes3D %matplotlib inline ###variable declarations nx = 41 ny = 41 nt = 120 c = 1 dx = 2 / (nx - 1) dy = 2 / (ny - 1) sigma = .0009 nu = 0.01 dt = sigma * dx * dy / nu x = numpy.linspace(0, 2, nx) y = numpy.linspace(0, 2, ny) u = numpy.ones((ny, nx)) # create a 1xn vector of 1's v = numpy.ones((ny, nx)) un = numpy.ones((ny, nx)) vn = numpy.ones((ny, nx)) comb = numpy.ones((ny, nx)) ###Assign initial conditions ##set hat function I.C. : u(.5<=x<=1 && .5<=y<=1 ) is 2 u[int(.5 / dy):int(1 / dy + 1),int(.5 / dx):int(1 / dx + 1)] = 2 ##set hat function I.C. : u(.5<=x<=1 && .5<=y<=1 ) is 2 v[int(.5 / dy):int(1 / dy + 1),int(.5 / dx):int(1 / dx + 1)] = 2 ###(plot ICs) for n in range(nt + 1): ##loop across number of time steps un = u.copy() vn = v.copy() u[1:-1, 1:-1] = (un[1:-1, 1:-1] - dt / dx * un[1:-1, 1:-1] * (un[1:-1, 1:-1] - un[1:-1, 0:-2]) - dt / dy * vn[1:-1, 1:-1] * (un[1:-1, 1:-1] - un[0:-2, 1:-1]) + nu * dt / dx**2 * (un[1:-1,2:] - 2 * un[1:-1, 1:-1] + un[1:-1, 0:-2]) + nu * dt / dy**2 * (un[2:, 1:-1] - 2 * un[1:-1, 1:-1] + un[0:-2, 1:-1])) v[1:-1, 1:-1] = (vn[1:-1, 1:-1] - dt / dx * un[1:-1, 1:-1] * (vn[1:-1, 1:-1] - vn[1:-1, 0:-2]) - dt / dy * vn[1:-1, 1:-1] * (vn[1:-1, 1:-1] - vn[0:-2, 1:-1]) + nu * dt / dx**2 * (vn[1:-1, 2:] - 2 * vn[1:-1, 1:-1] + vn[1:-1, 0:-2]) + nu * dt / dy**2 * (vn[2:, 1:-1] - 2 * vn[1:-1, 1:-1] + vn[0:-2, 1:-1])) u[0, :] = 1 u[-1, :] = 1 u[:, 0] = 1 u[:, -1] = 1 v[0, :] = 1 v[-1, :] = 1 v[:, 0] = 1 v[:, -1] = 1 fig = pyplot.figure(figsize=(11, 7), dpi=100) ax = fig.gca(projection='3d') X, Y = numpy.meshgrid(x, y) ax.plot_surface(X, Y, u, cmap=cm.viridis, rstride=1, cstride=1) ax.plot_surface(X, Y, v, cmap=cm.viridis, rstride=1, cstride=1) ax.set_xlabel('$x$') ax.set_ylabel('$y$'); ``` {{< figure src="https://blog.kakaocdn.net/dn/PL2CD/btq9fcyC1VV/MC1B8I2YedaaCFr5Lr06KK/img.png" >}} ## 2D Laplace Equation $$ \frac{\partial ^2 p}{\partial x^2} + \frac{\partial ^2 p}{\partial y^2} = 0 $$$$ p_{i,j}^n = \frac{\Delta y^2(p_{i+1,j}^n+p_{i-1,j}^n)+\Delta x^2(p_{i,j+1}^n + p_{i,j-1}^n)}{2(\Delta x^2 + \Delta y^2)} $$ ```python import numpy from matplotlib import pyplot, cm from mpl_toolkits.mplot3d import Axes3D %matplotlib inline def plot2D(x, y, p): fig = pyplot.figure(figsize=(11, 7), dpi=100) ax = fig.gca(projection='3d') X, Y = numpy.meshgrid(x, y) surf = ax.plot_surface(X, Y, p[:], rstride=1, cstride=1, cmap=cm.viridis, linewidth=0, antialiased=False) ax.set_xlim(0, 2) ax.set_ylim(0, 1) ax.view_init(30, 225) ax.set_xlabel('$x$') ax.set_ylabel('$y$') def laplace2d(p, y, dx, dy, l1norm_target): l1norm = 1 pn = numpy.empty_like(p) while l1norm > l1norm_target: pn = p.copy() p[1:-1, 1:-1] = ((dy**2 * (pn[1:-1, 2:] + pn[1:-1, 0:-2]) + dx**2 * (pn[2:, 1:-1] + pn[0:-2, 1:-1])) / (2 * (dx**2 + dy**2))) p[:, 0] = 0 # p = 0 @ x = 0 p[:, -1] = y # p = y @ x = 2 p[0, :] = p[1, :] # dp/dy = 0 @ y = 0 p[-1, :] = p[-2, :] # dp/dy = 0 @ y = 1 l1norm = (numpy.sum(numpy.abs(p[:]) - numpy.abs(pn[:])) / numpy.sum(numpy.abs(pn[:]))) return p nx = 31 ny = 31 c = 1 dx = 2 / (nx - 1) dy = 2 / (ny - 1) ##initial conditions p = numpy.zeros((ny, nx)) # create a XxY vector of 0's ##plotting aids x = numpy.linspace(0, 2, nx) y = numpy.linspace(0, 1, ny) ##boundary conditions p[:, 0] = 0 # p = 0 @ x = 0 p[:, -1] = y # p = y @ x = 2 p[0, :] = p[1, :] # dp/dy = 0 @ y = 0 p[-1, :] = p[-2, :] # dp/dy = 0 @ y = 1 p = laplace2d(p, y, dx, dy, 1e-4) plot2D(x, y, p) ``` {{< figure src="https://blog.kakaocdn.net/dn/bxvdGX/btq9goyjEC1/YkjgRDKkIZuAe2isKNsv60/img.png" >}} ## 2D Poisson Equation $$ \frac{\partial ^2 p}{\partial x^2} + \frac{\partial ^2 p}{\partial y^2} = b $$$$ p_{i,j}^{n}=\frac{(p_{i+1,j}^{n}+p_{i-1,j}^{n})\Delta y^2+(p_{i,j+1}^{n}+p_{i,j-1}^{n})\Delta x^2-b_{i,j}^{n}\Delta x^2\Delta y^2}{2(\Delta x^2+\Delta y^2)} $$ ```python import numpy from matplotlib import pyplot, cm from mpl_toolkits.mplot3d import Axes3D %matplotlib inline # Parameters nx = 50 ny = 50 nt = 100 xmin = 0 xmax = 2 ymin = 0 ymax = 1 dx = (xmax - xmin) / (nx - 1) dy = (ymax - ymin) / (ny - 1) # Initialization p = numpy.zeros((ny, nx)) pd = numpy.zeros((ny, nx)) b = numpy.zeros((ny, nx)) x = numpy.linspace(xmin, xmax, nx) y = numpy.linspace(xmin, xmax, ny) # Source b[int(ny / 4), int(nx / 4)] = 100 b[int(3 * ny / 4), int(3 * nx / 4)] = -100 for it in range(nt): pd = p.copy() p[1:-1,1:-1] = (((pd[1:-1, 2:] + pd[1:-1, :-2]) * dy**2 + (pd[2:, 1:-1] + pd[:-2, 1:-1]) * dx**2 - b[1:-1, 1:-1] * dx**2 * dy**2) / (2 * (dx**2 + dy**2))) p[0, :] = 0 p[ny-1, :] = 0 p[:, 0] = 0 p[:, nx-1] = 0 def plot2D(x, y, p): fig = pyplot.figure(figsize=(11, 7), dpi=100) ax = fig.gca(projection='3d') X, Y = numpy.meshgrid(x, y) surf = ax.plot_surface(X, Y, p[:], rstride=1, cstride=1, cmap=cm.viridis, linewidth=0, antialiased=False) ax.view_init(30, 225) ax.set_xlabel('$x$') ax.set_ylabel('$y$') plot2D(x, y, p) ``` ## ## Cavity Flow with Navier–Stokes $$ \frac{\partial \vec{v}}{\partial t}+(\vec{v}\cdot\nabla)\vec{v}=-\frac{1}{\rho}\nabla p + \nu \nabla^2\vec{v} $$$$ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} = -\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu \left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} \right) $$$$ \frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2} = -\rho\left(\frac{\partial u}{\partial x}\frac{\partial u}{\partial x}+2\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}+\frac{\partial v}{\partial y}\frac{\partial v}{\partial y} \right) $$$$ \begin{split}p_{i,j}^{n} = & \frac{\left(p_{i+1,j}^{n}+p_{i-1,j}^{n}\right) \Delta y^2 + \left(p_{i,j+1}^{n}+p_{i,j-1}^{n}\right) \Delta x^2}{2\left(\Delta x^2+\Delta y^2\right)} \\& -\frac{\rho\Delta x^2\Delta y^2}{2\left(\Delta x^2+\Delta y^2\right)} \\& \times \left[\frac{1}{\Delta t}\left(\frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}+\frac{v_{i,j+1}-v_{i,j-1}}{2\Delta y}\right)-\frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}\frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x} -2\frac{u_{i,j+1}-u_{i,j-1}}{2\Delta y}\frac{v_{i+1,j}-v_{i-1,j}}{2\Delta x}-\frac{v_{i,j+1}-v_{i,j-1}}{2\Delta y}\frac{v_{i,j+1}-v_{i,j-1}}{2\Delta y}\right]\end{split} $$ ```python import numpy from matplotlib import pyplot, cm from mpl_toolkits.mplot3d import Axes3D %matplotlib inline nx = 41 ny = 41 nt = 500 nit = 50 c = 1 dx = 2 / (nx - 1) dy = 2 / (ny - 1) x = numpy.linspace(0, 2, nx) y = numpy.linspace(0, 2, ny) X, Y = numpy.meshgrid(x, y) rho = 1 nu = .1 dt = .001 u = numpy.zeros((ny, nx)) v = numpy.zeros((ny, nx)) p = numpy.zeros((ny, nx)) b = numpy.zeros((ny, nx)) def build_up_b(b, rho, dt, u, v, dx, dy): b[1:-1, 1:-1] = (rho * (1 / dt * ((u[1:-1, 2:] - u[1:-1, 0:-2]) / (2 * dx) + (v[2:, 1:-1] - v[0:-2, 1:-1]) / (2 * dy)) - ((u[1:-1, 2:] - u[1:-1, 0:-2]) / (2 * dx))**2 - 2 * ((u[2:, 1:-1] - u[0:-2, 1:-1]) / (2 * dy) * (v[1:-1, 2:] - v[1:-1, 0:-2]) / (2 * dx))- ((v[2:, 1:-1] - v[0:-2, 1:-1]) / (2 * dy))**2)) return b def pressure_poisson(p, dx, dy, b): pn = numpy.empty_like(p) pn = p.copy() for q in range(nit): pn = p.copy() p[1:-1, 1:-1] = (((pn[1:-1, 2:] + pn[1:-1, 0:-2]) * dy**2 + (pn[2:, 1:-1] + pn[0:-2, 1:-1]) * dx**2) / (2 * (dx**2 + dy**2)) - dx**2 * dy**2 / (2 * (dx**2 + dy**2)) * b[1:-1,1:-1]) p[:, -1] = p[:, -2] # dp/dx = 0 at x = 2 p[0, :] = p[1, :] # dp/dy = 0 at y = 0 p[:, 0] = p[:, 1] # dp/dx = 0 at x = 0 p[-1, :] = 0 # p = 0 at y = 2 return p def cavity_flow(nt, u, v, dt, dx, dy, p, rho, nu): un = numpy.empty_like(u) vn = numpy.empty_like(v) b = numpy.zeros((ny, nx)) for n in range(nt): un = u.copy() vn = v.copy() b = build_up_b(b, rho, dt, u, v, dx, dy) p = pressure_poisson(p, dx, dy, b) u[1:-1, 1:-1] = (un[1:-1, 1:-1]- un[1:-1, 1:-1] * dt / dx * (un[1:-1, 1:-1] - un[1:-1, 0:-2]) - vn[1:-1, 1:-1] * dt / dy * (un[1:-1, 1:-1] - un[0:-2, 1:-1]) - dt / (2 * rho * dx) * (p[1:-1, 2:] - p[1:-1, 0:-2]) + nu * (dt / dx**2 * (un[1:-1, 2:] - 2 * un[1:-1, 1:-1] + un[1:-1, 0:-2]) + dt / dy**2 * (un[2:, 1:-1] - 2 * un[1:-1, 1:-1] + un[0:-2, 1:-1]))) v[1:-1,1:-1] = (vn[1:-1, 1:-1] - un[1:-1, 1:-1] * dt / dx * (vn[1:-1, 1:-1] - vn[1:-1, 0:-2]) - vn[1:-1, 1:-1] * dt / dy * (vn[1:-1, 1:-1] - vn[0:-2, 1:-1]) - dt / (2 * rho * dy) * (p[2:, 1:-1] - p[0:-2, 1:-1]) + nu * (dt / dx**2 * (vn[1:-1, 2:] - 2 * vn[1:-1, 1:-1] + vn[1:-1, 0:-2]) + dt / dy**2 * (vn[2:, 1:-1] - 2 * vn[1:-1, 1:-1] + vn[0:-2, 1:-1]))) u[0, :] = 0 u[:, 0] = 0 u[:, -1] = 0 u[-1, :] = 1 # set velocity on cavity lid equal to 1 v[0, :] = 0 v[-1, :] = 0 v[:, 0] = 0 v[:, -1] = 0 return u, v, p u = numpy.zeros((ny, nx)) v = numpy.zeros((ny, nx)) p = numpy.zeros((ny, nx)) b = numpy.zeros((ny, nx)) nt = 100 u, v, p = cavity_flow(nt, u, v, dt, dx, dy, p, rho, nu) fig = pyplot.figure(figsize=(11,7), dpi=100) # plotting the pressure field as a contour pyplot.contourf(X, Y, p, alpha=0.5, cmap=cm.viridis) pyplot.colorbar() # plotting the pressure field outlines pyplot.contour(X, Y, p, cmap=cm.viridis) # plotting velocity field pyplot.quiver(X[::2, ::2], Y[::2, ::2], u[::2, ::2], v[::2, ::2]) pyplot.xlabel('X') pyplot.ylabel('Y'); ``` {{< figure src="https://blog.kakaocdn.net/dn/VgtZK/btq9flvNFsA/4d03urU7VcLPRqzS5g40m1/img.png" >}} ```python u = numpy.zeros((ny, nx)) v = numpy.zeros((ny, nx)) p = numpy.zeros((ny, nx)) b = numpy.zeros((ny, nx)) nt = 700 u, v, p = cavity_flow(nt, u, v, dt, dx, dy, p, rho, nu) fig = pyplot.figure(figsize=(11, 7), dpi=100) pyplot.contourf(X, Y, p, alpha=0.5, cmap=cm.viridis) pyplot.colorbar() pyplot.contour(X, Y, p, cmap=cm.viridis) pyplot.quiver(X[::2, ::2], Y[::2, ::2], u[::2, ::2], v[::2, ::2]) pyplot.xlabel('X') pyplot.ylabel('Y'); ``` {{< figure src="https://blog.kakaocdn.net/dn/boYcRB/btq9fmasDP2/6w1UPPVU4mG7frDcjQtBIK/img.png" >}} ```python fig = pyplot.figure(figsize=(11, 7), dpi=100) pyplot.contourf(X, Y, p, alpha=0.5, cmap=cm.viridis) pyplot.colorbar() pyplot.contour(X, Y, p, cmap=cm.viridis) pyplot.streamplot(X, Y, u, v) pyplot.xlabel('X') pyplot.ylabel('Y'); ``` {{< figure src="https://blog.kakaocdn.net/dn/0J2aJ/btq9e57PmFB/UqArsnX9hzJ84H4rW5AtB1/img.png" >}} ## Channel Flow with Navier–Stokes $$ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)+F $$$$ \frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}=-\rho\left(\frac{\partial u}{\partial x}\frac{\partial u}{\partial x}+2\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}+\frac{\partial v}{\partial y}\frac{\partial v}{\partial y}\right) $$$$ \begin{split}p_{i,j}^{n} = & \frac{\left(p_{i+1,j}^{n}+p_{i-1,j}^{n}\right) \Delta y^2 + \left(p_{i,j+1}^{n}+p_{i,j-1}^{n}\right) \Delta x^2}{2(\Delta x^2+\Delta y^2)} \\& -\frac{\rho\Delta x^2\Delta y^2}{2\left(\Delta x^2+\Delta y^2\right)} \\& \times \left[\frac{1}{\Delta t} \left(\frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x} + \frac{v_{i,j+1}-v_{i,j-1}}{2\Delta y}\right) - \frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}\frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x} - 2\frac{u_{i,j+1}-u_{i,j-1}}{2\Delta y}\frac{v_{i+1,j}-v_{i-1,j}}{2\Delta x} - \frac{v_{i,j+1}-v_{i,j-1}}{2\Delta y}\frac{v_{i,j+1}-v_{i,j-1}}{2\Delta y}\right]\end{split} $$ ```python import numpy from matplotlib import pyplot, cm from mpl_toolkits.mplot3d import Axes3D %matplotlib inline def build_up_b(rho, dt, dx, dy, u, v): b = numpy.zeros_like(u) b[1:-1, 1:-1] = (rho * (1 / dt * ((u[1:-1, 2:] - u[1:-1, 0:-2]) / (2 * dx) + (v[2:, 1:-1] - v[0:-2, 1:-1]) / (2 * dy)) - ((u[1:-1, 2:] - u[1:-1, 0:-2]) / (2 * dx))**2 - 2 * ((u[2:, 1:-1] - u[0:-2, 1:-1]) / (2 * dy) * (v[1:-1, 2:] - v[1:-1, 0:-2]) / (2 * dx))- ((v[2:, 1:-1] - v[0:-2, 1:-1]) / (2 * dy))**2)) # Periodic BC Pressure @ x = 2 b[1:-1, -1] = (rho * (1 / dt * ((u[1:-1, 0] - u[1:-1,-2]) / (2 * dx) + (v[2:, -1] - v[0:-2, -1]) / (2 * dy)) - ((u[1:-1, 0] - u[1:-1, -2]) / (2 * dx))**2 - 2 * ((u[2:, -1] - u[0:-2, -1]) / (2 * dy) * (v[1:-1, 0] - v[1:-1, -2]) / (2 * dx)) - ((v[2:, -1] - v[0:-2, -1]) / (2 * dy))**2)) # Periodic BC Pressure @ x = 0 b[1:-1, 0] = (rho * (1 / dt * ((u[1:-1, 1] - u[1:-1, -1]) / (2 * dx) + (v[2:, 0] - v[0:-2, 0]) / (2 * dy)) - ((u[1:-1, 1] - u[1:-1, -1]) / (2 * dx))**2 - 2 * ((u[2:, 0] - u[0:-2, 0]) / (2 * dy) * (v[1:-1, 1] - v[1:-1, -1]) / (2 * dx))- ((v[2:, 0] - v[0:-2, 0]) / (2 * dy))**2)) return b def pressure_poisson_periodic(p, dx, dy): pn = numpy.empty_like(p) for q in range(nit): pn = p.copy() p[1:-1, 1:-1] = (((pn[1:-1, 2:] + pn[1:-1, 0:-2]) * dy**2 + (pn[2:, 1:-1] + pn[0:-2, 1:-1]) * dx**2) / (2 * (dx**2 + dy**2)) - dx**2 * dy**2 / (2 * (dx**2 + dy**2)) * b[1:-1, 1:-1]) # Periodic BC Pressure @ x = 2 p[1:-1, -1] = (((pn[1:-1, 0] + pn[1:-1, -2])* dy**2 + (pn[2:, -1] + pn[0:-2, -1]) * dx**2) / (2 * (dx**2 + dy**2)) - dx**2 * dy**2 / (2 * (dx**2 + dy**2)) * b[1:-1, -1]) # Periodic BC Pressure @ x = 0 p[1:-1, 0] = (((pn[1:-1, 1] + pn[1:-1, -1])* dy**2 + (pn[2:, 0] + pn[0:-2, 0]) * dx**2) / (2 * (dx**2 + dy**2)) - dx**2 * dy**2 / (2 * (dx**2 + dy**2)) * b[1:-1, 0]) # Wall boundary conditions, pressure p[-1, :] =p[-2, :] # dp/dy = 0 at y = 2 p[0, :] = p[1, :] # dp/dy = 0 at y = 0 return p ##variable declarations nx = 41 ny = 41 nt = 10 nit = 50 c = 1 dx = 2 / (nx - 1) dy = 2 / (ny - 1) x = numpy.linspace(0, 2, nx) y = numpy.linspace(0, 2, ny) X, Y = numpy.meshgrid(x, y) ##physical variables rho = 1 nu = .1 F = 1 dt = .01 #initial conditions u = numpy.zeros((ny, nx)) un = numpy.zeros((ny, nx)) v = numpy.zeros((ny, nx)) vn = numpy.zeros((ny, nx)) p = numpy.ones((ny, nx)) pn = numpy.ones((ny, nx)) b = numpy.zeros((ny, nx)) udiff = 1 stepcount = 0 while udiff > .001: un = u.copy() vn = v.copy() b = build_up_b(rho, dt, dx, dy, u, v) p = pressure_poisson_periodic(p, dx, dy) u[1:-1, 1:-1] = (un[1:-1, 1:-1] - un[1:-1, 1:-1] * dt / dx * (un[1:-1, 1:-1] - un[1:-1, 0:-2]) - vn[1:-1, 1:-1] * dt / dy * (un[1:-1, 1:-1] - un[0:-2, 1:-1]) - dt / (2 * rho * dx) * (p[1:-1, 2:] - p[1:-1, 0:-2]) + nu * (dt / dx**2 * (un[1:-1, 2:] - 2 * un[1:-1, 1:-1] + un[1:-1, 0:-2]) + dt / dy**2 * (un[2:, 1:-1] - 2 * un[1:-1, 1:-1] + un[0:-2, 1:-1])) + F * dt) v[1:-1, 1:-1] = (vn[1:-1, 1:-1] - un[1:-1, 1:-1] * dt / dx * (vn[1:-1, 1:-1] - vn[1:-1, 0:-2]) - vn[1:-1, 1:-1] * dt / dy * (vn[1:-1, 1:-1] - vn[0:-2, 1:-1]) - dt / (2 * rho * dy) * (p[2:, 1:-1] - p[0:-2, 1:-1]) + nu * (dt / dx**2 * (vn[1:-1, 2:] - 2 * vn[1:-1, 1:-1] + vn[1:-1, 0:-2]) + dt / dy**2 * (vn[2:, 1:-1] - 2 * vn[1:-1, 1:-1] + vn[0:-2, 1:-1]))) # Periodic BC u @ x = 2 u[1:-1, -1] = (un[1:-1, -1] - un[1:-1, -1] * dt / dx * (un[1:-1, -1] - un[1:-1, -2]) - vn[1:-1, -1] * dt / dy * (un[1:-1, -1] - un[0:-2, -1]) - dt / (2 * rho * dx) * (p[1:-1, 0] - p[1:-1, -2]) + nu * (dt / dx**2 * (un[1:-1, 0] - 2 * un[1:-1,-1] + un[1:-1, -2]) + dt / dy**2 * (un[2:, -1] - 2 * un[1:-1, -1] + un[0:-2, -1])) + F * dt) # Periodic BC u @ x = 0 u[1:-1, 0] = (un[1:-1, 0] - un[1:-1, 0] * dt / dx * (un[1:-1, 0] - un[1:-1, -1]) - vn[1:-1, 0] * dt / dy * (un[1:-1, 0] - un[0:-2, 0]) - dt / (2 * rho * dx) * (p[1:-1, 1] - p[1:-1, -1]) + nu * (dt / dx**2 * (un[1:-1, 1] - 2 * un[1:-1, 0] + un[1:-1, -1]) + dt / dy**2 * (un[2:, 0] - 2 * un[1:-1, 0] + un[0:-2, 0])) + F * dt) # Periodic BC v @ x = 2 v[1:-1, -1] = (vn[1:-1, -1] - un[1:-1, -1] * dt / dx * (vn[1:-1, -1] - vn[1:-1, -2]) - vn[1:-1, -1] * dt / dy * (vn[1:-1, -1] - vn[0:-2, -1]) - dt / (2 * rho * dy) * (p[2:, -1] - p[0:-2, -1]) + nu * (dt / dx**2 * (vn[1:-1, 0] - 2 * vn[1:-1, -1] + vn[1:-1, -2]) + dt / dy**2 * (vn[2:, -1] - 2 * vn[1:-1, -1] + vn[0:-2, -1]))) # Periodic BC v @ x = 0 v[1:-1, 0] = (vn[1:-1, 0] - un[1:-1, 0] * dt / dx * (vn[1:-1, 0] - vn[1:-1, -1]) - vn[1:-1, 0] * dt / dy * (vn[1:-1, 0] - vn[0:-2, 0]) - dt / (2 * rho * dy) * (p[2:, 0] - p[0:-2, 0]) + nu * (dt / dx**2 * (vn[1:-1, 1] - 2 * vn[1:-1, 0] + vn[1:-1, -1]) + dt / dy**2 * (vn[2:, 0] - 2 * vn[1:-1, 0] + vn[0:-2, 0]))) # Wall BC: u,v = 0 @ y = 0,2 u[0, :] = 0 u[-1, :] = 0 v[0, :] = 0 v[-1, :]=0 udiff = (numpy.sum(u) - numpy.sum(un)) / numpy.sum(u) stepcount += 1 fig = pyplot.figure(figsize = (11,7), dpi=100) pyplot.quiver(X[::3, ::3], Y[::3, ::3], u[::3, ::3], v[::3, ::3]); ``` {{< figure src="https://blog.kakaocdn.net/dn/du6hla/btq9fdKZP6o/ifKi67Tsr8khMmReNSHn5K/img.png" >}} 출처> [CFD Python: 12 steps to Navier-Stokes :: Lorena A. Barba Group (lorenabarba.com)](https://lorenabarba.com/blog/cfd-python-12-steps-to-navier-stokes/)