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<title>Direct numerical simulation</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Direct numerical simulation</span></span>

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<p>A <b>direct numerical simulation</b> (<b>DNS</b>)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is a <a href="Simulation" title="Simulation">simulation</a> in <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a> (CFD) in which the <a href="Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> are numerically solved without any <a href="Turbulence_model" class="mw-redirect" title="Turbulence model">turbulence model</a>. This means that the whole range of <a href="Three-dimensional_space" title="Three-dimensional space">spatial</a> and <a href="Time" title="Time">temporal</a> scales of the <a href="Turbulence" title="Turbulence">turbulence</a> must be resolved. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales (<a href="Kolmogorov_microscales" title="Kolmogorov microscales">Kolmogorov microscales</a>), up to the <a href="Integral_length_scale" title="Integral length scale">integral scale</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span>, associated with the motions containing most of the kinetic energy. The Kolmogorov scale, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

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</math></span><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span>, is given by

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<annotation encoding="application/x-tex">{\displaystyle \eta =(\nu ^{3}/\varepsilon )^{1/4}}</annotation>

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</math></span><img src="./a59d61fbb48c8cdb113ceeccdc51365e17243397.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.333ex; height:3.343ex;" alt="{\displaystyle \eta =(\nu ^{3}/\varepsilon )^{1/4}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

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</math></span><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"></span> is the <a href="Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

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</math></span><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"></span> is the rate of <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a> dissipation. On the other hand, the integral scale depends usually on the <a href="Spatial_scale" title="Spatial scale">spatial scale</a> of the boundary conditions.

</p><p>To satisfy these resolution requirements, the number of points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> along a given mesh direction with increments <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span>, must be

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<annotation encoding="application/x-tex">{\displaystyle Nh>L,\,}</annotation>

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</math></span><img src="./1f87215dcc410e09d947d6057df8a5cfa2c052e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.118ex; height:2.509ex;" alt="{\displaystyle Nh>L,\,}" loading="lazy"></span></dd></dl>

<p>so that the integral scale is contained within the computational domain, and also

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<annotation encoding="application/x-tex">{\displaystyle h\leq \eta ,\,}</annotation>

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</math></span><img src="./3bbcbdb18acc7cdd6c9ff16aa435acd1a3134e86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.641ex; height:2.676ex;" alt="{\displaystyle h\leq \eta ,\,}" loading="lazy"></span></dd></dl>

<p>so that the Kolmogorov scale can be resolved.

</p><p>Since

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon \approx {u'}^{3}/L,}</annotation>

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</math></span><img src="./dd2d572a9a62a0bee5eab2d9a2779bd3475e7627.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.643ex; height:3.509ex;" alt="{\displaystyle \varepsilon \approx {u'}^{3}/L,}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u'}">

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<annotation encoding="application/x-tex">{\displaystyle u'}</annotation>

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</math></span><img src="./bee645f272e64333868e2baa275419eca4ee0718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle u'}" loading="lazy"></span> is the <a href="Root_mean_square" title="Root mean square">root mean square</a> (RMS) of the <a href="Velocity" title="Velocity">velocity</a>, the previous relations imply that a three-dimensional DNS requires a number of mesh points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle N^{3}}</annotation>

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</math></span><img src="./0c55dcbd0d1893c037f846f52adda4f714b3f7d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.177ex; height:2.676ex;" alt="{\displaystyle N^{3}}" loading="lazy"></span> satisfying

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<annotation encoding="application/x-tex">{\displaystyle N^{3}\geq \mathrm {Re} ^{9/4}=\mathrm {Re} ^{2.25}}</annotation>

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</math></span><img src="./fdab9c730a820a1b9c034fa2f1f891a122890498.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.714ex; height:3.009ex;" alt="{\displaystyle N^{3}\geq \mathrm {Re} ^{9/4}=\mathrm {Re} ^{2.25}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} }</annotation>

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</math></span><img src="./ce9a978f729af1a34028e60d45f484f0908e4be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Re} }" loading="lazy"></span> is the turbulent <a href="Reynolds_number" title="Reynolds number">Reynolds number</a>:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {u'L}{\nu }}.}</annotation>

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</math></span><img src="./6724bda9b98fc56aca309c15a854da015d12d033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.922ex; height:5.509ex;" alt="{\displaystyle \mathrm {Re} ={\frac {u'L}{\nu }}.}" loading="lazy"></span></dd></dl>

<p>Hence, the memory storage requirement in a DNS grows very fast with the Reynolds number. In addition, given the very large memory necessary, the integration of the solution in time must be done by an explicit method. This means that in order to be accurate, the integration, for most <a href="Discretization" title="Discretization">discretization</a> methods, must be done with a time step, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>

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</math></span><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy"></span>, small enough such that a fluid particle moves only a fraction of the mesh spacing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> in each step. That is,

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<annotation encoding="application/x-tex">{\displaystyle C={\frac {u'\Delta t}{h}}<1}</annotation>

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</math></span><img src="./847d47354b0d0b98f712acb8269d51ef5d3beaa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.752ex; height:5.676ex;" alt="{\displaystyle C={\frac {u'\Delta t}{h}}<1}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> is here the <a href="Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition" title="Courant–Friedrichs–Lewy condition">Courant number</a>). The total time interval simulated is generally proportional to the turbulence time scale <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math></span><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"></span> given by

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<annotation encoding="application/x-tex">{\displaystyle \tau ={\frac {L}{u'}}.}</annotation>

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</math></span><img src="./1c4dc7431bd1ec879983991607ff0b17421ec7cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.798ex; height:5.176ex;" alt="{\displaystyle \tau ={\frac {L}{u'}}.}" loading="lazy"></span></dd></dl>

<p>Combining these relations, and the fact that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> must be of the order of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

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</math></span><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span>, the number of time-integration steps must be proportional to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/(C\eta )}">

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<annotation encoding="application/x-tex">{\displaystyle L/(C\eta )}</annotation>

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</math></span><img src="./f41528fb7b31ca1249818d1c7952a97d98bd4e38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.49ex; height:2.843ex;" alt="{\displaystyle L/(C\eta )}" loading="lazy"></span>. On the other hand, from the definitions for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} }</annotation>

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</math></span><img src="./ce9a978f729af1a34028e60d45f484f0908e4be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Re} }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

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</math></span><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> given above, it follows that

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<annotation encoding="application/x-tex">{\displaystyle {\frac {L}{\eta }}\sim \mathrm {Re} ^{3/4},}</annotation>

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</math></span><img src="./09f65fb86de069728b0ad444ebb68d12d84d85d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.605ex; height:5.676ex;" alt="{\displaystyle {\frac {L}{\eta }}\sim \mathrm {Re} ^{3/4},}" loading="lazy"></span></dd></dl>

<p>and consequently, the number of time steps grows also as a <a href="Power_law" title="Power law">power law</a> of the Reynolds number.

</p><p>One can estimate that the number of floating-point operations required to complete the simulation is proportional to the number of mesh points and the number of time steps, and in conclusion, the number of operations grows as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ^{3}}</annotation>

</semantics>

</math></span><img src="./2f795f5fc8fc60a155c4900319a3b516430f8283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.797ex; height:2.676ex;" alt="{\displaystyle \mathrm {Re} ^{3}}" loading="lazy"></span>.

</p><p>Therefore, the computational cost of DNS is very high, even at low Reynolds numbers. For the Reynolds numbers encountered in most industrial applications, the computational resources required by a DNS would exceed the capacity of the <a href="Supercomputer" title="Supercomputer">most powerful computers currently available</a>. However, direct numerical simulation is a useful tool in fundamental research in turbulence. Using DNS it is possible to perform "numerical experiments", and extract from them information difficult or impossible to obtain in the laboratory, allowing a better understanding of the physics of turbulence. Also, direct numerical simulations are useful in the development of turbulence models for practical applications, such as sub-grid scale models for <a href="Large_eddy_simulation" title="Large eddy simulation">large eddy simulation</a> (LES) and models for methods that solve the <a href="Reynolds-averaged_Navier%E2%80%93Stokes_equations" title="Reynolds-averaged Navier–Stokes equations">Reynolds-averaged Navier–Stokes equations</a> (RANS). This is done by means of "a priori" tests, in which the input data for the model is taken from a DNS simulation, or by "a posteriori" tests, in which the results produced by the model are compared with those obtained by DNS.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Here the origin of the term <i>direct numerical simulation</i> (see e.g. p. 385 in <style data-mw-deduplicate="TemplateStyles:r1238218222">

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFOrszag1970" class="citation journal cs1"><a href="Steven_A._Orszag" class="mw-redirect" title="Steven A. Orszag">Orszag, Steven A.</a> (1970). "Analytical Theories of Turbulence". <i><a href="Journal_of_Fluid_Mechanics" title="Journal of Fluid Mechanics">Journal of Fluid Mechanics</a></i>. <b>41</b> (1970): <span class="nowrap">363–</span>386. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1970JFM....41..363O">1970JFM....41..363O</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112070000642">10.1017/S0022112070000642</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122834319">122834319</a>.</cite>) owes to the fact that, at that time, there were considered to be just two principal ways of getting <i>theoretical</i> results regarding turbulence, namely via turbulence theories (like the direct interaction approximation) and <i>directly</i> from solution of the Navier–Stokes equations.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://eprints.soton.ac.uk/66182/1/A_primer_on_DNS.pdf">https://eprints.soton.ac.uk/66182/1/A_primer_on_DNS.pdf</a> "A Primer on Direct Numerical Simulation of Turbulence –

Methods, Procedures and Guidelines", Coleman and Sandberg, 2010</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

This article is issued from <a class="external text" title="Last edited on 2025-03-19" href="https://en.wikipedia.org/wiki/?title=Direct_numerical_simulation&oldid=1281270612">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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