{"id":13,"date":"2022-01-02T15:55:04","date_gmt":"2022-01-02T07:55:04","guid":{"rendered":"https:\/\/zhangxinzhuo.club\/?p=13"},"modified":"2022-08-11T10:48:50","modified_gmt":"2022-08-11T02:48:50","slug":"n","status":"publish","type":"post","link":"https:\/\/blog.zhangxinzhuo.com\/?p=13","title":{"rendered":"CFD\u5b66\u4e60\u7b14\u8bb01\u2014Navier-Stokes \u65b9\u7a0b\u7684\u57fa\u672c\u63a8\u5bfc"},"content":{"rendered":"\n<p>\u5728\u5bf9CFD\u5373\u8ba1\u7b97\u6d41\u4f53\u529b\u5b66\u7684\u5b66\u4e60\u8fc7\u7a0b\u4e2d\uff0cNavier-Stokes\u65b9\u7a0b\u662f\u7ed5\u4e0d\u8fc7\u53bb\u7684\u3002\u5728\u8f83\u5e38\u89c1\u7684\u7269\u7406\u573a\u4e2d\uff0cNavier-Stokes \u65b9\u7a0b\u8fd8\u662f\u9002\u7528\u7684\u3002\u4e0b\u9762\u4e3b\u8981\u6574\u7406\u8bb0\u5f55\u4e00\u4e0bNavier-Stokes\u65b9\u7a0b\u7684\u57fa\u672c\u63a8\u5bfc\u8fc7\u7a0b\u3002<\/p>\n\n\n\n<p>\\[\\frac{\\partial \\rho}{\\partial t}+\\nabla \\cdot(\\rho \\mathbf{U})=0\\]<\/p>\n\n\n\n<p>\\begin{equation}\\frac{\\partial \\rho \\mathbf{U}}{\\partial t}+\\nabla \\cdot(\\rho \\mathbf{U U})=-\\nabla p+\\nabla \\cdot \\tau\\end{equation}<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u57fa\u7840\u77e5\u8bc6<\/h2>\n\n\n\n<p>\u5728\u5bf9Navier-Stokes\u65b9\u7a0b\u8fdb\u884c\u63a8\u5bfc\u4e4b\u524d\uff0c\u8fd8\u662f\u6709\u5fc5\u8981\u660e\u786e\u76f8\u5173\u6570\u5b66\u6982\u5ff5\uff0c\u6bd5\u7adf\u5bf9\u4e8e\u4e00\u4e2a\u516b\u767e\u5e74\u6ca1\u7528\u8fc7\u9ad8\u6570\u7684\u4eba\u6765\u8bf4\uff0c\u5355\u9760\u56de\u5fc6\u6015\u662f\u4e0d\u600e\u4e48\u9760\u8c31\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"Taylor-series-expansion\">\u6cf0\u52d2\u5c55\u5f00<\/h3>\n\n\n\n<p>\u6cf0\u52d2\u5c55\u5f00\u5f0f\u5e94\u8be5\u5728\u5927\u4e00\u9ad8\u6570\u6709\u8bb2\u8fc7\uff0c\u7b80\u5355\u6765\u8bf4\u5c31\u662f\u4e00\u4e2a\u7528\u51fd\u6570\u5728\u67d0\u70b9\u7684\u4fe1\u606f\u63cf\u8ff0\u5176\u9644\u8fd1\u53d6\u503c\u7684\u516c\u5f0f\u3002\u65e2\u53ef\u7528\u4e8e\u4e00\u5143\u51fd\u6570\u4e5f\u53ef\u7528\u4e8e\u591a\u5143\u51fd\u6570\uff0c\u4e0b\u5217\u516c\u5f0f\u6458\u81ea<a href=\"https:\/\/zh.wikipedia.org\/wiki\/%E6%B3%B0%E5%8B%92%E5%85%AC%E5%BC%8F\">\u7ef4\u57fa\u767e\u79d1<\/a>\uff0c\u662f\u5e26\u6709\u62c9\u683c\u6717\u65e5\u578b\u4f59\u9879\u7684\u6cf0\u52d2\u516c\u5f0f\uff1a<\/p>\n\n\n\n<p>\\begin{equation}<br>f(x)=f(a)+\\frac{f^{\\prime}(a)}{1 !}(x-a)+\\frac{f^{(2)}(a)}{2 !}(x-a)^{2}+\\cdots+\\frac{f^{(n)}(a)}{n !}(x-a)^{n}+\\frac{f^{(n+1)}(\\theta)}{(n+1) !}(x-a)^{(n+1)}<br>\\end{equation}<\/p>\n\n\n\n<p>\u4e00\u822c\u6765\u8bf4\uff0c\u5728\u4e0b\u8ff0Navier-Stokes\u65b9\u7a0b\u7684\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\uff0c\u53ea\u7528\u5230\u4e86\u4e00\u9636\u8868\u8fbe\u5f62\u5f0f\u3002\u81f3\u4e8e\u4e3a\u4ec0\u4e48\u4e0d\u7528\u5230\u9ad8\u9636\u7684\u5f62\u5f0f\uff0c\u4e2a\u4eba\u6000\u7591\u662f\u56e0\u4e3a\u9488\u5bf9\u6709\u9650\u4f53\u79ef\u6cd5\u6765\u8bf4\uff0c\u4e00\u9636\u5f62\u5f0f\u7684\u7cbe\u5ea6\u53ef\u80fd\u5c31\u8db3\u591f\u4e86\uff1f\u4e0b\u9762\u5206\u522b\u662f\u4e00\u5143\u53ca\u591a\u5143\u51fd\u6570\u7684\u6cf0\u52d2\u5c55\u5f00\u5f0f\uff1a<\/p>\n\n\n\n<p>\u4e00\u5143\u6cf0\u52d2\u5c55\u5f00\u5f0f\uff1a<\/p>\n\n\n\n<p>\\begin{equation}f\\left(x_{1}\\right)=f\\left(x_{0}\\right)+\\left.\\frac{\\partial f(x)}{\\partial x}\\right|_{x=x_{0}}\\left(x_{1}-x_{0}\\right)+\\ldots\\end{equation}<\/p>\n\n\n\n<p>\u591a\u5143\u6cf0\u52d2\u5c55\u5f00\u5f0f\uff1a<\/p>\n\n\n\n<p>\\begin{equation}\\begin{array}{r}f(x, y, z, t)=f\\left(x_{1}, y_{1}, z_{1}, t_{1}\\right)+\\left.\\frac{\\partial f(x, y, z, t)}{\\partial x}\\right|_{x=x_{1}}\\left(x-x_{1}\\right)+\\left.\\frac{\\partial f(x, y, z, t)}{\\partial y}\\right|_{y=y_{1}}\\left(y-y_{1}\\right)+ \\\\\\left.\\frac{\\partial f(x, y, z, t)}{\\partial z}\\right|_{z=z_{1}}\\left(z-z_{1}\\right)+\\left.\\frac{\\partial f(x, y, z, t)}{\\partial t}\\right|_{t=t_{1}}\\left(t-t_{1}\\right)+\\ldots\\end{array}\\end{equation}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6570\u5b66\u7b26\u53f7<\/h3>\n\n\n\n<p>\u663e\u7136\uff0c\u5728\u6587\u7ae0\u5f00\u5934\u6240\u5c55\u793a\u7684Navier-Stokes\u65b9\u7a0b\u6709\u4e00\u70b9\u6666\u6da9\uff0c\u4e3b\u8981\u662f\u56e0\u4e3a\u5b83\u7684\u8868\u8fbe\u5f62\u5f0f\u4e0a\u5f15\u5165\u4e86\u6563\u5ea6\u3001\u68af\u5ea6\u7b49\u5185\u5bb9\uff0c\u5982\u679c\u5c06\u5176\u66ff\u6362\u4e3a\u5e38\u89c1\u7684\u5fae\u5206\u5f62\u5f0f\uff0c\u76f8\u4fe1\u8d77\u7801\u80fd\u8ba9\u4eba\u6709\u4e00\u70b9\u4eb2\u8fd1\u611f\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Nabla\u7b97\u5b50<\/h4>\n\n\n\n<p>\u9996\u5148\u5f15\u5165Nabla\u7b97\u5b50\uff0c\u4e5f\u79f0\u4e3a\u5012\u4e09\u89d2\u7b97\u5b50\uff0c\u8fd9\u53ef\u80fd\u662f\u56e0\u4e3a\u5b83\u7684\u5199\u6cd5\u3002\u5176\u5f62\u5f0f\u4e3a\uff1a<\/p>\n\n\n\n<p>\\begin{equation}\\nabla=\\frac{d}{d r}\\end{equation}<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u68af\u5ea6<\/h4>\n\n\n\n<p>\u53ef\u76f4\u63a5\u4f7f\u7528Nabla\u7b97\u5b50\u8868\u8fbe\u68af\u5ea6\uff0c\u5f53\u5176\u76f4\u63a5\u4f5c\u7528\u4e8e\u51fd\u6570\u65f6\uff0c\u65e0\u8bba\u5176\u662f\u5426\u4e3a\u77e2\u91cf\u6216\u662f\u6807\u91cf\uff0c\u90fd\u8868\u793a\u5176\u68af\u5ea6\u3002\u6807\u91cf\u51fd\u6570\u7684\u68af\u5ea6\u4e3a\u5411\u91cf\uff0c\u5411\u91cf\u7684<a href=\"https:\/\/zh.wikipedia.org\/wiki\/%E6%A2%AF%E5%BA%A6\">\u68af\u5ea6<\/a>\u4e3a\u4e8c\u9636\u5f20\u91cf\u3002<\/p>\n\n\n\n<p>\\begin{equation}\\nabla f=\\left(\\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z}\\right)=\\frac{\\partial f}{\\partial x} \\mathbf{i}+\\frac{\\partial f}{\\partial y} \\mathbf{j}+\\frac{\\partial f}{\\partial z} \\mathbf{k}\\end{equation}<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u6563\u5ea6<\/h4>\n\n\n\n<p>\u6563\u5ea6\u662f\u5411\u91cf\u573a\u7684\u4e00\u79cd\u5f3a\u5ea6\u6027\u8d28\uff0c\u5c31\u5982\u540c\u5bc6\u5ea6\u3001\u6d53\u5ea6\u3001\u6e29\u5ea6\u4e00\u6837\uff0c\u5b83\u5bf9\u5e94\u7684\u5e7f\u5ef6\u6027\u8d28\u662f\u4e00\u4e2a\u5c01\u95ed\u533a\u57df\u8868\u9762\u7684\u901a\u91cf\uff0c\u6240\u4ee5\u8bf4\u6563\u5ea6\u662f\u901a\u91cf\u7684\u4f53\u5bc6\u5ea6\u3002\u6d41\u4f53\u529b\u5b66\u4e2d\uff0c\u6563\u5ea6\u4e3a\u96f6\u7684\u6d41\u4f53\u79f0\u4e3a\u4e0d\u53ef\u538b\u7f29\u6d41\u4f53\uff0c\u4e5f\u5c31\u662f\u8bf4\u6b64\u6d41\u4f53\u4e2d\u4e0d\u4f1a\u6709\u4e00\u90e8\u5206\u51ed\u7a7a\u6d88\u5931\u6216\u7a81\u7136\u4ea7\u751f\uff0c\u6bcf\u4e2a\u5fae\u5c0f\u65f6\u95f4\u95f4\u9694\u4e2d\u6d41\u5165\u4e00\u4e2a\u5fae\u5c0f\u4f53\u5143\u7684\u6d41\u4f53\u603b\u91cf\u90fd\u7b49\u4e8e\u5728\u6b64\u65f6\u95f4\u95f4\u9694\u5185\u6d41\u51fa\u6b64\u4f53\u5143\u7684\u6d41\u4f53\u603b\u91cf\u3002<\/p>\n\n\n\n<p>$$\\nabla \\cdot \\mathbf{A}=\\frac{\\partial A_{x}}{\\partial x}+\\frac{\\partial A_{y}}{\\partial y}+\\frac{\\partial A_{z}}{\\partial z}$$<\/p>\n\n\n\n<p>\u5b9e\u9645\u4e0a\uff0c\u6563\u5ea6\u662f\u4e00\u4e2a\u6807\u91cf\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u65cb\u5ea6<\/h4>\n\n\n\n<p>\u65cb\u5ea6\u662f\u5411\u91cf\u573a\u7684\u4e00\u79cd\u5f3a\u5ea6\u6027\u8d28\uff0c\u5c31\u5982\u540c\u5bc6\u5ea6\u3001\u6d53\u5ea6\u3001\u6e29\u5ea6\u4e00\u6837\uff0c\u5b83\u5bf9\u5e94\u7684\u5e7f\u5ef6\u6027\u8d28\u662f\u5411\u91cf\u573a\u6cbf\u4e00\u4e2a\u95ed\u5408\u66f2\u7ebf\u7684\u73af\u91cf\uff0c\u6240\u4ee5\u8bf4\u65cb\u5ea6\u662f\u73af\u91cf\u7684\u9762\u5bc6\u5ea6\u3002\u5982\u679c\u4e00\u4e2a\u5411\u91cf\u573a\u4e2d\u5904\u5904\u7684\u65cb\u5ea6\u90fd\u662f\u96f6\uff0c\u5219\u79f0\u8fd9\u4e2a\u573a\u4e3a\u65e0\u65cb\u573a\u6216\u4fdd\u5b88\u573a\u3002<\/p>\n\n\n\n<p>\u4e0d\u540c\u4e8e\u68af\u5ea6\u548c\u6563\u5ea6\uff0c\u65cb\u5ea6\u4e0d\u80fd\u7b80\u5355\u7684\u63a8\u5e7f\u5230\u5176\u4ed6\u7ef4\u5ea6\uff1b\u67d0\u4e9b\u63a8\u5e7f\u662f\u53ef\u80fd\u7684\uff0c\u4f46\u662f\u53ea\u6709\u5728\u4e09\u7ef4\u4e2d\uff0c\u5728\u51e0\u4f55\u4e0a\u5b9a\u4e49\u7684\u5411\u91cf\u573a\u65cb\u5ea6\u8fd8\u662f\u5411\u91cf\u573a\u3002<\/p>\n\n\n\n<p>\\begin{equation}\\boldsymbol{\\nabla} \\times \\mathbf{A}=\\left(\\frac{\\partial A_{z}}{\\partial y}-\\frac{\\partial A_{y}}{\\partial z}\\right) \\mathbf{i}+\\left(\\frac{\\partial A_{x}}{\\partial z}-\\frac{\\partial A_{z}}{\\partial x}\\right) \\mathbf{j}+\\left(\\frac{\\partial A_{y}}{\\partial x}-\\frac{\\partial A_{x}}{\\partial y}\\right) \\mathbf{k}\\end{equation}<\/p>\n\n\n\n<p>\u5f53\u7136\uff0c\u4e5f\u53ef\u8868\u793a\u4e3a\u884c\u5217\u5f0f\u3002<\/p>\n\n\n\n<p>\\begin{equation}\\left|\\begin{array}{ccc}\\mathbf{i} &amp; \\mathbf{j} &amp; \\mathbf{k} \\\\\\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\A_{x} &amp; A_{y} &amp; A_{z}\\end{array}\\right|\\end{equation}<\/p>\n\n\n\n<p>\u4e0d\u8fc7\u4eca\u5929\u7684\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\uff0c\u5012\u6ca1\u6709\u6d89\u53ca\u5230\u65cb\u5ea6\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">N-S\u65b9\u7a0b<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u524d\u8a00<\/h3>\n\n\n\n<p>\u6d41\u4f53\u4e0e\u56fa\u4f53\u6709\u4e00\u4e2a\u660e\u663e\u7684\u533a\u522b\uff0c\u5c31\u662f\u5728\u627f\u53d7\u526a\u5e94\u529b\u65f6\u5c06\u4f1a\u53d1\u751f\u8fde\u7eed\u53d8\u5f62\u3002\u5f53\u6d41\u4f53\u53d7\u5230\u526a\u5e94\u529b\u53d8\u5f62\u6216\u62c9\u4f38\u5e94\u529b\u65f6\uff0c\u5176\u4f1a\u4ea7\u751f\u4e00\u5b9a\u963b\u529b\uff0c\u8fd9\u6837\u7684\u963b\u529b\u88ab\u79f0\u4e3a\u7c98\u6ede\u529b\u6216\u8005\u7c98\u6027\u529b\u3002\u6211\u4eec\u7528\u9ecf\u5ea6\uff08<em>Viscosity<\/em>\uff09\u6765\u8868\u5f81\u6d41\u4f53\u7684\u9ecf\u6027\uff0c\u9ecf\u5ea6\u4e5f\u88ab\u79f0\u4e3a\u8fd0\u52a8\u7c98\u5ea6\u3001\u7c98\u6ede\u7cfb\u6570\u7b49\u3002\u5f53\u4e00\u6d41\u4f53\u9ecf\u5ea6\u6052\u5b9a\u65f6\uff0c\u79f0\u5176\u4e3a\u725b\u987f\u6d41\u4f53\uff0c\u5f53\u4e00\u6d41\u4f53\u9ecf\u5ea6\u4e0d\u6052\u5b9a\u65f6\uff0c\u79f0\u5176\u4e3a\u975e\u725b\u987f\u6d41\u4f53\u3002<\/p>\n\n\n\n<p>\u6d41\u4f53\u4e3b\u8981\u5305\u62ec\u6db2\u4f53\u548c\u6c14\u4f53\uff0c\u81f3\u4e8e\u7b49\u79bb\u5b50\u4f53\u6682\u4e0d\u505a\u8ba8\u8bba\u3002\u6db2\u4f53\u4e0e\u6c14\u4f53\u76f8\u6bd4\uff0c\u5176\u53ef\u538b\u7f29\u6027\u5927\u5927\u964d\u4f4e\u3002\u901a\u5e38\u6765\u770b\uff0c\u5c06\u6db2\u4f53\u89c6\u4e3a\u4e0d\u53ef\u538b\u7f29\u6d41\u4f53\uff0c\u5176\u5bc6\u5ea6\u4e3a\u5e38\u6570\u3002\u81f3\u4e8e\u662f\u5426\u5c06\u6c14\u4f53\u89c6\u4e3a\u53ef\u538b\u7f29\u6d41\u4f53\uff0c\u8fd9\u4e2a\u9700\u8981\u7ed3\u5408\u5b9e\u9645\u5206\u6790\u60c5\u51b5\u3002<\/p>\n\n\n\n<p>\u540c\u65f6\uff0c\u5728\u4f7f\u7528N-S\u65b9\u7a0b\u7684\u4e00\u4e2a\u91cd\u8981\u524d\u63d0\u662f\u8fde\u7eed\u4ecb\u8d28\u5047\u5b9a\uff08<em>the concept of continuum<\/em>\uff09\u3002\u6b64\u65f6\uff0c\u4e00\u4e2a\u5355\u5143\u4f53\u5185\u6709\u8db3\u591f\u7684\u6d41\u4f53\u5206\u5b50\uff0c\u5728\u5b8f\u89c2\u4e0a\u5145\u5206\u5c0f\uff0c\u800c\u5728\u5fae\u89c2\u4e0a\u5145\u5206\u5927\u3002<\/p>\n\n\n\n<p>\u603b\u4f53\u800c\u8a00\uff0c\u4e3a\u63cf\u8ff0\u6d41\u4f53\u7684\u8fd0\u52a8\uff0c\u6211\u4eec\u5c06\u4f7f\u7528\u4ee5\u4e0b\u4e09\u79cd\u65b9\u7a0b\u3002\u5176\u4e00\u662f\u8fde\u7eed\u6027\u65b9\u7a0b\uff0c\u6ee1\u8db3\u8d28\u91cf\u5b88\u6052\u5b9a\u5f8b\uff1b\u5176\u4e8c\u662f\u52a8\u91cf\u65b9\u7a0b\uff0c\u6ee1\u8db3\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\uff1b\u5728\u8003\u8651\u4f20\u70ed\u7684\u60c5\u51b5\u4e0b\uff0c\u8fd8\u5e94\u9644\u52a0\u80fd\u91cf\u65b9\u7a0b\uff0c\u6ee1\u8db3\u80fd\u91cf\u5b88\u6052\u5b9a\u5f8b\u3002<\/p>\n\n\n\n<p>\u591a\u6570\u60c5\u51b5\u4e0b\uff0cCFD\u6c42\u89e3\u65b9\u6cd5\u4e3a\u6709\u9650\u4f53\u79ef\u6cd5\uff08<em>FVM<\/em>\uff09\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6d41\u52a8\u6a21\u578b<\/h3>\n\n\n\n<p>\u5728\u8ba8\u8bbaN-S\u65b9\u7a0b\u524d\uff0c\u6211\u4eec\u8981\u660e\u786e\u6d41\u52a8\u6a21\u578b\u3002<\/p>\n\n\n\n<p>\u6d41\u52a8\u6a21\u578b\u5206\u4e3a\u6709\u9650\u63a7\u5236\u4f53\u6a21\u578b\u548c\u65e0\u7a77\u5c0f\u5fae\u56e2\u6a21\u578b\uff0c\u540c\u65f6\uff0c\u4e5f\u53ef\u8fdb\u4e00\u6b65\u5206\u4e3a\u7a7a\u95f4\u4f4d\u7f6e\u56fa\u5b9a\u548c\u968f\u6d41\u7ebf\u8fd0\u52a8\uff0c\u5bf9\u5e94\u7684\u63cf\u8ff0\u4e3a\u6b27\u62c9\u63cf\u8ff0\u548c\u62c9\u683c\u6717\u65e5\u63cf\u8ff0\u3002\u5728\u672c\u6587\u4e2d\uff0c\u4e3a\u65b9\u4fbf\u63a8\u5bfc\uff0c\u4f7f\u7528\u7684\u662f\u7a7a\u95f4\u4f4d\u7f6e\u56fa\u5b9a\u7684\u65e0\u7a77\u5c0f\u5fae\u56e2\u6a21\u578b\u3002\u6b64\u65f6\uff0c\u8d28\u91cf\u3001\u4f53\u79ef\u7684\u63cf\u8ff0\u53ef\u4f7f\u7528d<em>V<\/em>\u3001d<em>m<\/em>\u6765\u8868\u793a\u3002<\/p>\n\n\n\n<p>\u5e76\u5f15\u5165\u7269\u8d28\u5bfc\u6570\u7684\u6982\u5ff5\u3002\u63cf\u8ff0t<sub>1<\/sub>\u65f6\u523b\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e0b\u7684\u4e00\u5fae\u56e2\u901f\u5ea6<em>U<sub>1<\/sub>(x<sub>1<\/sub>,y<sub>1<\/sub>,z<sub>1<\/sub>,t<sub>1<\/sub>)<\/em>\uff0c\u968f\u5fae\u56e2\u8fd0\u52a8\uff0c t<sub>2<\/sub>\u65f6\u523b\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e0b\u7684\u4e00\u5fae\u56e2\u6e29\u5ea6<em>U<sub>2<\/sub>(x<sub>2<\/sub>,y<sub>2<\/sub>,z<sub>2<\/sub>,t<sub>2<\/sub>)<\/em> \u3002\u57fa\u4e8e<a href=\"#Taylor-series-expansion\">\u591a\u5143\u51fd\u6570\u7684\u6cf0\u52d2\u5c55\u5f00<\/a>\uff0c\u663e\u7136\u6709\u5982\u4e0b\u65b9\u7a0b\u3002<\/p>\n\n\n\n<p>\\begin{equation}\\begin{aligned}U_{2} &amp;=U_{1}+\\left.\\frac{\\partial T}{\\partial x}\\right|_{x=x_{1}}\\left(x_{2}-x_{1}\\right)+\\left.\\frac{\\partial T}{\\partial y}\\right|_{y=y_{1}}\\left(y_{2}-y_{1}\\right)+\\left.\\frac{\\partial T}{\\partial z}\\right|_{z=z_{1}}\\left(z_{2}-z_{1}\\right)+\\left.\\frac{\\partial T}{\\partial t}\\right|_{t=t_{1}}\\left(t_{2}-t_{1}\\right)\\end{aligned}\\end{equation}<\/p>\n\n\n\n<p>\u4e24\u4fa7\u5206\u522b\u9664\u4ee5(<em>t<sub>2<\/sub><\/em>&#8211; <em>t<sub>1<\/sub><\/em>)\uff1a<\/p>\n\n\n\n<p>\\begin{equation}\\frac{u_{2}-u_{1}}{t_{2}-t_{1}}=\\left.\\frac{\\partial U}{\\partial x}\\right|_{x=x_{1}} \\frac{x_{2}-x_{1}}{t_{2}-t_{1}}+\\left.\\frac{\\partial U}{\\partial y}\\right|_{y=y_{1}} \\frac{y_{2}-y_{1}}{t_{2}-t_{1}}+\\left.\\frac{\\partial U}{\\partial z}\\right|_{z=z_{1}} \\frac{z_{1}-z_{1}}{t_{x}-t_{1}}+\\left.\\frac{\\partial U}{\\partial t}\\right|_{t=t_{1}} \\end{equation}<\/p>\n\n\n\n<p>\u5f53 <em>t<sub>2<\/sub><\/em>\u65e0\u9650\u63a5\u8fd1 <em>t<sub>1<\/sub><\/em>\u65f6\uff0c\u6839\u636e\u901f\u5ea6\u7684\u5b9a\u4e49\u5373\u53ef\u5f97\u5230\u901f\u5ea6U \u5728\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e0b\u7684\u7269\u8d28\u5bfc\u6570\u7684\u5b9a\u4e49 \uff0c<\/p>\n\n\n\n<p>\\begin{equation}\\frac{d U}{d t}=\\frac{\\partial U}{\\partial x} \\cdot u+\\frac{\\partial U}{\\partial y} \\cdot v+\\frac{\\partial U}{\\partial t} \\cdot w+\\frac{\\partial U}{\\partial t}\\end{equation}<\/p>\n\n\n\n<p>\u5176\u4f59\u6027\u8d28\uff0c\u4f8b\u5982\u7279\u5f81\u7ebf\u7b49\uff0c\u7559\u5f85\u4e0b\u6b21\u8bb0\u5f55\u3002<\/p>\n\n\n\n<p>\\begin{equation}\\underbrace{\\frac{\\mathrm{D}}{\\mathrm{D} t}}_{\\text {Lagrangian }}=\\underbrace{\\frac{\\partial}{\\partial t}+\\mathrm{U} \\cdot \\nabla}_{\\text {Euler }}\\end{equation}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u8fde\u7eed\u6027\u65b9\u7a0b<\/h3>\n\n\n\n<p>\u4ee5\u8f83\u4e3a\u7b80\u5355\u7684\u6b27\u62c9\u6cd5\u5bf9\u8fde\u7eed\u6027\u65b9\u7a0b\u8fdb\u884c\u63a8\u5bfc\uff0c\u6b27\u62c9\u89c2\u70b9\u4e0b\u7684\u8d28\u91cf\u5b88\u6052\u610f\u5473\u7740\u4f4d\u7f6e\u56fa\u5b9a\u65e0\u7a77\u5c0f\u5fae\u56e2\u8d28\u91cf\u7684\u53d8\u5316= \u6d41\u5165\u65e0\u7a77\u5c0f\u5fae\u56e2\u8d28\u91cf\u2212\u6d41\u51fa\u65e0\u7a77\u5c0f\u5fae\u56e2\u8d28\u91cf\u3002\u4e5f\u5c31\u662f\u8fd9\u4e24\u8005\u53d8\u5316\u7387\u76f8\u540c\u3002<\/p>\n\n\n\n<p>\u9996\u5148\u8868\u8fbe\u65e0\u7a77\u5c0f\u5fae\u56e2\u8d28\u91cf\u7684\u53d8\u5316\u7387\uff0c\u663e\u7136\u6d41\u4f53\u5fae\u56e2\u4f53\u79ef\u4e3a\\(\\mathrm{d} V=\\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z\\)\uff0c\u57fa\u4e8e\u5bc6\u5ea6\\(\\rho\\)\uff0c\u53ef\u5f97\u5230\u5176\u5bf9\u5e94\u7684\u65e0\u7a77\u5c0f\u5fae\u56e2\u8d28\u91cf\uff1a<\/p>\n\n\n\n<p>\\[\\mathrm{d} m=\\rho \\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z\\]<\/p>\n\n\n\n<p>\u5176\u53d8\u5316\u7387\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[\\frac{\\partial \\mathrm{d} m}{\\partial t}=\\frac{\\partial \\rho \\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z}{\\partial t}=\\frac{\\partial \\rho}{\\partial t} \\mathrm{~d} x \\mathrm{~d} y \\mathrm{~d} z\\]<\/p>\n\n\n\n<p>\u968f\u540e\u8868\u8fbe\uff08\u6d41\u5165\u65e0\u7a77\u5c0f\u5fae\u56e2\u8d28\u91cf\u2212\u6d41\u51fa\u65e0\u7a77\u5c0f\u5fae\u56e2\u8d28\u91cf\uff09\u7684\u53d8\u5316\u7387\uff0c\u5176\u4e2d\u4ee5\\(x\\)\u65b9\u5411\u4e3a\u4f8b\uff0c\u6b64\u65f6\u5355\u4f4d\u65f6\u95f4\u5185\u7684\u6d41\u5165\u8d28\u91cf\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[\\rho u \\mathrm{~d} y \\mathrm{~d} z\\]<\/p>\n\n\n\n<p>\u6839\u636e\u6cf0\u52d2\u516c\u5f0f\u7684\u4e00\u9636\u5c55\u5f00\uff0c\u80fd\u5f97\u5230\u5355\u4f4d\u65f6\u95f4\u5185\u7684\u6d41\u51fa\u8d28\u91cf\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[\\left(\\rho u+\\frac{\\partial \\rho u}{\\partial x} \\mathrm{~d} x\\right) \\mathrm{d} y \\mathrm{~d} z\\]<\/p>\n\n\n\n<p>\u56e0\u6b64\uff0c\u53ef\u5f97\u5230\uff08\u6d41\u5165\u65e0\u7a77\u5c0f\u5fae\u56e2\u8d28\u91cf\u2212\u6d41\u51fa\u65e0\u7a77\u5c0f\u5fae\u56e2\u8d28\u91cf\uff09\u7684\u53d8\u5316\u7387\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[\\rho u \\mathrm{~d} y \\mathrm{~d} z-\\left(\\rho u+\\frac{\\partial \\rho u}{\\partial x} \\mathrm{~d} x\\right) \\mathrm{d} y \\mathrm{~d} z=-\\frac{\\partial \\rho u}{\\partial x} \\mathrm{~d} x \\mathrm{~d} y \\mathrm{~d} z\\]<\/p>\n\n\n\n<p>\u540c\u7406\uff0c \\(y\\)\u65b9\u5411\u4e0a\u7684\u8d28\u91cf\u53d8\u5316\u7387\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[ -\\frac{\\partial \\rho v}{\\partial y} \\mathrm{~d} x \\mathrm{~d} y \\mathrm{~d} z\\] <\/p>\n\n\n\n<p>\\(z\\)\u65b9\u5411\u4e0a\u7684\u8d28\u91cf\u53d8\u5316\u7387\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[ -\\frac{\\partial \\rho w}{\\partial z} \\mathrm{~d} x \\mathrm{~d} y \\mathrm{~d} z\\] <\/p>\n\n\n\n<p>\u6b64\u65f6\uff0c\u8054\u7acb\u4e0a\u8ff0\u65b9\u7a0b\uff0c\u6709<\/p>\n\n\n\n<p>\\[\\frac{\\partial \\rho}{\\partial t} \\mathrm{~d} x \\mathrm{~d} y \\mathrm{~d} z=-\\left(\\frac{\\partial \\rho u}{\\partial x}+\\frac{\\partial \\rho v}{\\partial y}+\\frac{\\partial \\rho w}{\\partial z}\\right) \\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z\\]<\/p>\n\n\n\n<p>\u4e5f\u5373\u4e3a\u8fde\u7eed\u65b9\u7a0b\uff1a<\/p>\n\n\n\n<p>\\[\\frac{\\partial \\rho}{\\partial t}+\\frac{\\partial \\rho u}{\\partial x}+\\frac{\\partial \\rho v}{\\partial y}+\\frac{\\partial \\rho w}{\\partial z}=0\\]<\/p>\n\n\n\n<p>\u57fa\u4e8e\u6563\u5ea6\uff0c\u5f97\u5230\u4e0b\u5f0f\uff1a<\/p>\n\n\n\n<p>\\begin{equation}\\frac{\\partial \\rho}{\\partial t}+\\nabla \\cdot(\\rho \\mathbf{U})=0\\end{equation}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u52a8\u91cf\u65b9\u7a0b<\/h3>\n\n\n\n<p>\u52a8\u91cf\u65b9\u7a0b\u7684\u63a8\u5bfc\u57fa\u4e8e\u7a7a\u95f4\u4f4d\u7f6e\u79fb\u52a8\u7684\u65e0\u7a77\u5c0f\u5fae\u56e2\uff0c\u8fd9\u662f\u56e0\u4e3a\u5176\u6bd4\u8f83\u7b80\u5355\u3002<\/p>\n\n\n\n<p>\u9996\u5148\u5bf9\u65e0\u7a77\u5c0f\u5fae\u56e2\u8fdb\u884c\u53d7\u529b\u5206\u6790\u3002\u5176\u53d7\u529b\u53ef\u5206\u4e3a\u4f53\u79ef\u529b\u548c\u8868\u9762\u529b\u3002\u4f53\u79ef\u529b\u4f5c\u7528\u5728\u65e0\u7a77\u5c0f\u5fae\u56e2\u7684\u5168\u90e8\u4f53\u79ef\u4e0a\uff0c\u4f8b\u5982\u91cd\u529b\u3001\u7535\u78c1\u529b\uff1b\u8868\u9762\u529b\u4f5c\u7528\u5728\u65e0\u7a77\u5c0f\u5fae\u56e2\u7684\u9762\u4e0a\uff0c\u5305\u62ec\u538b\u529b\u3001\u8868\u9762\u5f20\u529b\u3002\u5728\u5f53\u524d\u5206\u6790\u63a8\u5bfc\u4e2d\uff0c\u6700\u91cd\u8981\u7684\u8868\u9762\u529b\u662f\u538b\u529b\u548c\u5e94\u529b\uff0c\u4e0d\u8003\u8651\u5176\u4ed6\u6e90\u9879\u529b\u7684\u60c5\u51b5\u4e0b\uff0c\u91cd\u529b\u662f\u6700\u91cd\u8981\u7684\u4f53\u79ef\u529b\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u538b\u529b<\/h4>\n\n\n\n<p>\u538b\u529b\u5728\u8fd9\u91cc\u6307\u9759\u538b\uff0c\u7528\\(p\\)\u6765\u8868\u793a\uff0c\u5176\u5f71\u54cd\u7684\u662f\u65e0\u7a77\u5c0f\u5fae\u56e2\u7684\u538b\u7f29\u4e0e\u81a8\u80c0\uff0c\u540c\u65f6\uff0c\u538b\u529b\u4f5c\u4e3a\u6b63\u5e94\u529b\uff0c\u5176\u6c38\u8fdc\u4f5c\u7528\u4e8e\u65e0\u7a77\u5c0f\u5fae\u56e2\u9762\u7684\u6cd5\u5411\u3002<\/p>\n\n\n\n<p>\u4ee5\\(x\\)\u65b9\u5411\u4e3a\u4f8b\uff0c\u5176\u5de6\u4fa7\u53d7\u538b\u529b\u4e3a\uff1a<\/p>\n\n\n\n<p> \\[ p\\mathrm{~d} y \\mathrm{~d} z\\] <\/p>\n\n\n\n<p> \u5176\u53f3\u4fa7\u53d7\u538b\u529b\u4e3a\uff1a <\/p>\n\n\n\n<p>\\[ &#8211; p\\mathrm{~d} y \\mathrm{~d} z -\\frac{\\partial p}{\\partial x} \\mathrm{~d} x \\mathrm{~d} y \\mathrm{~d} z\\] <\/p>\n\n\n\n<p>\u5219\u5728\\(x\\)\u65b9\u5411\u4e0a\uff0c\u53d7\u538b\u529b\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[ -\\frac{\\partial p}{\\partial x} \\mathrm{~d} x \\mathrm{~d} y \\mathrm{~d} z\\] <\/p>\n\n\n\n<p>\u540c\u7406\\(y\\)\u65b9\u5411\u4e0a\u53d7\u538b\u529b\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[ -\\frac{\\partial p}{\\partial y} \\mathrm{~d} x \\mathrm{~d} y \\mathrm{~d} z\\] <\/p>\n\n\n\n<p>\u540c\u7406\\(z\\)\u65b9\u5411\u4e0a\u53d7\u538b\u529b\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[ -\\frac{\\partial p}{\\partial z} \\mathrm{~d} x \\mathrm{~d} y \\mathrm{~d} z\\] <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u526a\u5207\u5e94\u529b<\/h4>\n\n\n\n<p>\u53c2\u8003<a href=\"http:\/\/blog.sina.com.cn\/s\/blog_83997b020102x4r7.html\">\u6587\u7ae0<\/a>\uff0c\u6682\u65f6\u4e0d\u6253\u51fa\u5177\u4f53\u516c\u5f0f\u53ca\u56fe\u89e3\uff0c\u76ee\u524d\u6839\u636e\u526a\u5207\u5e94\u529b\u6765\u770b\uff0c\u53ef\u7ed9\u51fa\u5404\u4e2a\u65b9\u5411\u4e0a\u7684\u53d7\u529b\u3002<\/p>\n\n\n\n<p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u65e0\u7a77\u5c0f\u5fae\u56e2\u53d7\u529b\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[\\mathbf{F}=\\left[\\begin{array}{l}<br>\\left(-\\frac{\\partial p}{\\partial x}+\\frac{\\partial \\tau_{x x}}{\\partial x}+\\frac{\\partial \\tau_{y x}}{\\partial y}+\\frac{\\partial \\tau_{z x}}{\\partial z}\\right) \\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z \\<br>\\left(-\\frac{\\partial p}{\\partial y}+\\frac{\\partial \\tau_{x y}}{\\partial x}+\\frac{\\partial \\tau_{y y}}{\\partial y}+\\frac{\\partial \\tau_{z y}}{\\partial z}\\right) \\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z \\<br>\\left(-\\frac{\\partial p}{\\partial z}+\\frac{\\partial \\tau_{x z}}{\\partial x}+\\frac{\\partial \\tau_{y z}}{\\partial y}+\\frac{\\partial \\tau_{z z}}{\\partial z}\\right) \\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z<br>\\end{array}\\right]\\]<\/p>\n\n\n\n<p>\u57fa\u4e8e\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\uff0c\u5df2\u77e5\u65e0\u7a77\u5c0f\u6d41\u4f53\u5fae\u56e2\u8d28\u91cf\u4e3a\\(\\mathrm{d} m=\\rho \\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z\\)\uff0c\u6d41\u4f53\u52a0\u901f\u5ea6\u4e3a\\(\\left[\\frac{\\mathrm{D} u}{\\mathrm{D} t}, \\frac{\\mathrm{D} v}{\\mathrm{D} t}, \\frac{\\mathrm{D} w}{\\mathrm{D} t}\\right]^{\\mathrm{T}}\\)\uff0c\u7ed3\u5408\u4e0a\u8ff0\u53d7\u529b\uff0c\u53ef\u5f97\u5230\u5982\u4e0b\u65b9\u7a0b\u3002<\/p>\n\n\n\n<p>\u4ee5\\(x\\)\u65b9\u5411\u4e3a\u4f8b\uff0c\u5728\\(x\\)\u65b9\u5411\u4e0a\u7684\u52a8\u91cf\u65b9\u7a0b\u4e3a\uff1a<\/p>\n\n\n\n<p>\\[\\rho \\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z \\frac{\\mathrm{D} u}{\\mathrm{D} t}=\\left(-\\frac{\\partial p}{\\partial x}+\\frac{\\partial \\tau_{x x}}{\\partial x}+\\frac{\\partial \\tau_{y x}}{\\partial y}+\\frac{\\partial \\tau_{z x}}{\\partial z}\\right) \\mathrm{d} x \\mathrm{~d} y \\mathrm{~d} z\\]<\/p>\n\n\n\n<p>\u6b64\u65f6\u6709\u5982\u4e0b\u65b9\u7a0b\uff1a<\/p>\n\n\n\n<p>\\[\\rho \\frac{\\mathrm{D} u}{\\mathrm{D} t}=-\\frac{\\partial p}{\\partial x}+\\frac{\\partial \\tau_{x x}}{\\partial x}+\\frac{\\partial \\tau_{y x}}{\\partial y}+\\frac{\\partial \\tau_{z x}}{\\partial z}\\] <\/p>\n\n\n\n<p><br>\\[ \\rho \\frac{\\mathrm{D} v}{\\mathrm{D} t}=-\\frac{\\partial p}{\\partial y}+\\frac{\\partial \\tau_{x y}}{\\partial x}+\\frac{\\partial \\tau_{y y}}{\\partial y}+\\frac{\\partial \\tau_{z y}}{\\partial z}\\] <\/p>\n\n\n\n<p><br>\\[ \\rho \\frac{\\mathrm{D} w}{\\mathrm{D} t}=-\\frac{\\partial p}{\\partial z}+\\frac{\\partial \\tau_{x z}}{\\partial x}+\\frac{\\partial \\tau_{y z}}{\\partial y}+\\frac{\\partial \\tau_{z z}}{\\partial z}\\]<\/p>\n\n\n\n<p>\u7ed3\u5408\u6563\u5ea6\u3001\u68af\u5ea6\u7b49\u5b9a\u4e49\uff0c\u65b9\u7a0b\u53ef\u5199\u4e3a\u5982\u4e0b\u5f62\u5f0f\uff1a<\/p>\n\n\n\n<p>\\[\\rho \\frac{\\mathrm{DU}}{\\mathrm{D} t}=-\\nabla p+\\nabla \\cdot \\tau\\]<\/p>\n\n\n\n<p>\u7ed3\u5408\u7269\u8d28\u5bfc\u6570\uff0c\u65b9\u7a0b\u53ef\u5199\u4e3a\u5982\u4e0b\u5f62\u5f0f\uff0c\u8fd9\u4e5f\u662f\u8f83\u4e3a\u5e38\u89c1\u7684\u8868\u8fbe\uff1a<\/p>\n\n\n\n<p>\\[\\rho\\left(\\frac{\\partial \\mathbf{U}}{\\partial t}+\\mathbf{U} \\cdot \\nabla \\mathbf{U}\\right)=-\\nabla p+\\nabla \\cdot \\tau\\]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u975e\u725b\u987f\u6d41\u4f53\u7684\u5f15\u5165<\/h3>\n\n\n\n<p>\u660e\u65e5\u518d\u5199\uff0c\u563f\u563f<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5728\u5bf9CFD\u5373\u8ba1\u7b97\u6d41\u4f53\u529b\u5b66\u7684\u5b66\u4e60\u8fc7\u7a0b\u4e2d\uff0cNavier-Stokes\u65b9\u7a0b\u662f\u7ed5\u4e0d\u8fc7\u53bb\u7684\u3002\u5728\u8f83\u5e38\u89c1\u7684\u7269\u7406\u573a\u4e2d\uff0cNavi [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-13","post","type-post","status-publish","format-standard","hentry","category-cfdtext"],"_links":{"self":[{"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=\/wp\/v2\/posts\/13","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=13"}],"version-history":[{"count":66,"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=\/wp\/v2\/posts\/13\/revisions"}],"predecessor-version":[{"id":84,"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=\/wp\/v2\/posts\/13\/revisions\/84"}],"wp:attachment":[{"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.zhangxinzhuo.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}