https://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&feed=atom&action=historyIntroduction to turbulence/Turbulence kinetic energy - Revision history2024-03-29T15:19:56ZRevision history for this page on the wikiMediaWiki 1.16.5https://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=22673&oldid=prevZonder at 12:47, 13 December 20132013-12-13T12:47:53Z<p></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>where the symmetric part is the strain<del class="diffchange diffchange-inline">-</del>rate tensor, <math> s_{ij} </math>, and the anti-symmetric part is the rotation<del class="diffchange diffchange-inline">-</del>rate tensor <math> \omega_{ij} </math>, defined by:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>where the symmetric part is the strain rate tensor, <math> s_{ij} </math>, and the anti-symmetric part is the rotation rate tensor <math> \omega_{ij} </math>, defined by:</div></td></tr>
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</table>Zonderhttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=14018&oldid=prevBluebase at 13:17, 21 March 20122012-03-21T13:17:16Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(8)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(8)</td></tr></table></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Both equations 6 and 8 play an important role in the study of turbulence. The first form given by equation 6 will provide the framework for understanding the dynamics of turbulent motion. The second form, equation 8 forms the basis for most of the second-order closure attempts at turbulence modelling; e.g., the socalled k-e models ( usually referred to as the “k-epsilon models”). This because it has fewer unknowns to be modelled, although this comes at the expense of some extra assumptions about the last term. It is only the last term in equation 6 that can be identified as the true rate of dissipation of turbulence kinetic energy, unlike the last term in equation 8 which is only the dissipation when the flow is ''homogeneous''. We will talk about <del class="diffchange diffchange-inline">homogeniety </del>below, but suffice it to say now that it never occurs in nature. Nonetheless, many flows can be assumed to be homogeneous ''at the scales of turbulence which are important to this term'', so-called ''local <del class="diffchange diffchange-inline">homogeniety</del>''.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Both equations 6 and 8 play an important role in the study of turbulence. The first form given by equation 6 will provide the framework for understanding the dynamics of turbulent motion. The second form, equation 8 forms the basis for most of the second-order closure attempts at turbulence modelling; e.g., the socalled k-e models ( usually referred to as the “k-epsilon models”). This because it has fewer unknowns to be modelled, although this comes at the expense of some extra assumptions about the last term. It is only the last term in equation 6 that can be identified as the true rate of dissipation of turbulence kinetic energy, unlike the last term in equation 8 which is only the dissipation when the flow is ''homogeneous''. We will talk about <ins class="diffchange diffchange-inline">homogeneity </ins>below, but suffice it to say now that it never occurs in nature. Nonetheless, many flows can be assumed to be homogeneous ''at the scales of turbulence which are important to this term'', so-called ''local <ins class="diffchange diffchange-inline">homogeneity</ins>''.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Each term in the equation for the kinetic energy of the turbulence has a distinct role to play in the overall kinetic energy balance. Briefly these are:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Each term in the equation for the kinetic energy of the turbulence has a distinct role to play in the overall kinetic energy balance. Briefly these are:</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(9)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(9)</td></tr></table></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* Rate of change of kinetic energy per unit mass due to convection (or advection) by the mean flow through an <del class="diffchange diffchange-inline">inhomogenous </del>field :</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* Rate of change of kinetic energy per unit mass due to convection (or advection) by the mean flow through an <ins class="diffchange diffchange-inline">inhomogeneous </ins>field :</div></td></tr>
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</table>Bluebasehttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=14017&oldid=prevBluebase at 13:13, 21 March 20122012-03-21T13:13:55Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This reduces to equation 14 only for a Newtonian fluid. In non-Newtonian fluids, protions of this product may not be negative implying that it may not all represent an irrecoverable loss of fluctuating kinetic energy.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This reduces to equation 14 only for a Newtonian fluid. In non-Newtonian fluids, protions of this product may not be negative implying that it may not all represent an irrecoverable loss of fluctuating kinetic energy.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>It will be shown in the following chapter on [[Introduction to turbulence/Stationarity and <del class="diffchange diffchange-inline">homogenity</del>|stationarity and <del class="diffchange diffchange-inline">homogenity</del>]] that the dissipation of turbulence energy mostly takes place at the smallest turbulence scales, and that those scales can be characterized by so-called Kolmogorov microscale defined by:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>It will be shown in the following chapter on [[Introduction to turbulence/Stationarity and <ins class="diffchange diffchange-inline">homogeneity</ins>|stationarity and <ins class="diffchange diffchange-inline">homogeneity</ins>]] that the dissipation of turbulence energy mostly takes place at the smallest turbulence scales, and that those scales can be characterized by so-called Kolmogorov microscale defined by:</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Almost always <math> l \propto L </math>, but the relation is at most only exact theoretically in the limit of infinite Reynolds number since the constant of proportionality is Reynolds number dependent. The Reynolds number dependence of the ratio <math> L/l </math> for grid turbulence is illustrated in <font color=orange>Figure 4.1</font>. Many interpret this data to suggest that this ratioapproaches a constant and ignore the scatter. In fact some assume ratio to be constant and even refer to <math> l </math> though it were the real integral scale. Others argue that the scatter is because of the differing upstream conditions and that the ratio may not be constant at all. It is really hard to tell who is right in the absence of facilities or simulations in which the Reynolds number can vary very much for fixed initial conditions. This all may leave you feeling a bit confused, but that’s the way turbulence is right now. It’s a lot easier to teach if we just tell you one view, but that’s not very good preparation for the future.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Almost always <math> l \propto L </math>, but the relation is at most only exact theoretically in the limit of infinite Reynolds number since the constant of proportionality is Reynolds number dependent. The Reynolds number dependence of the ratio <math> L/l </math> for grid turbulence is illustrated in <font color=orange>Figure 4.1</font>. Many interpret this data to suggest that this ratioapproaches a constant and ignore the scatter. In fact some assume ratio to be constant and even refer to <math> l </math> though it were the real integral scale. Others argue that the scatter is because of the differing upstream conditions and that the ratio may not be constant at all. It is really hard to tell who is right in the absence of facilities or simulations in which the Reynolds number can vary very much for fixed initial conditions. This all may leave you feeling a bit confused, but that’s the way turbulence is right now. It’s a lot easier to teach if we just tell you one view, but that’s not very good preparation for the future.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Here is what we can say for sure. Only the integral scale, <math>L</math>, is a physical length scale, meaning that it can be directly observed in the flow by spectral or correlation measurements (as shown in the following chapters on [[Introduction to turbulence/Stationarity and <del class="diffchange diffchange-inline">homogenity</del>|stationarity and <del class="diffchange diffchange-inline">homogenity</del>]] and [[Introduction to turbulence/<del class="diffchange diffchange-inline">Homogenous </del>turbulence|<del class="diffchange diffchange-inline">homogenous </del>turbulence]]). The pseudo-integral scale, <math>l</math>, on the other hand is simply a definition; and it is only at infinite turbulence Reynolds number that it may have physical significance. But it is certainly a useful</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Here is what we can say for sure. Only the integral scale, <math>L</math>, is a physical length scale, meaning that it can be directly observed in the flow by spectral or correlation measurements (as shown in the following chapters on [[Introduction to turbulence/Stationarity and <ins class="diffchange diffchange-inline">homogeneity</ins>|stationarity and <ins class="diffchange diffchange-inline">homogeneity</ins>]] and [[Introduction to turbulence/<ins class="diffchange diffchange-inline">Homogeneous </ins>turbulence|<ins class="diffchange diffchange-inline">homogeneous </ins>turbulence]]). The pseudo-integral scale, <math>l</math>, on the other hand is simply a definition; and it is only at infinite turbulence Reynolds number that it may have physical significance. But it is certainly a useful</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>approximation at large, but finite, Reynolds numbers. We will talk about these subtle but important distinctions later when we consider homogeneous flows, but it is especially important when considering similarity theories of turbulence. For</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>approximation at large, but finite, Reynolds numbers. We will talk about these subtle but important distinctions later when we consider homogeneous flows, but it is especially important when considering similarity theories of turbulence. For</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>now simply file away in your memory a note of caution about using equation 17 too freely. And do not be fooled by the cute description this provides. It is just that, a description, and not really an explanation of why all this happens — sort</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>now simply file away in your memory a note of caution about using equation 17 too freely. And do not be fooled by the cute description this provides. It is just that, a description, and not really an explanation of why all this happens — sort</div></td></tr>
</table>Bluebasehttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=13192&oldid=prevAlexskillen at 12:58, 24 August 20112011-08-24T12:58:57Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(5)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(5)</td></tr></table></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>By dividing equation 1 by 2 and inserting this definition, the equation for the average kinetic energy per unit mass of the fluctuating motion can be re-written as:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>By dividing equation 1 by 2<ins class="diffchange diffchange-inline">.0 </ins>and inserting this definition, the equation for the average kinetic energy per unit mass of the fluctuating motion can be re-written as:</div></td></tr>
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</table>Alexskillenhttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=13191&oldid=prevAlexskillen at 12:57, 24 August 20112011-08-24T12:57:50Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(5)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(5)</td></tr></table></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>By dividing equation 1 by <del class="diffchange diffchange-inline">equation </del>2 and inserting this definition, the equation for the average kinetic energy per unit mass of the fluctuating motion can be re-written as:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>By dividing equation 1 by 2 and inserting this definition, the equation for the average kinetic energy per unit mass of the fluctuating motion can be re-written as:</div></td></tr>
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</table>Alexskillenhttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=12726&oldid=prevMalkavian GT: /* Rate of dissipation of the turbulence kinetic energy */2011-03-20T18:46:51Z<p><span class="autocomment">Rate of dissipation of the turbulence kinetic energy</span></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\epsilon = 2\nu \left\langle s_{ij} s_{ij} \right\rangle = \nu \left\{ \left\langle \frac{\partial u_{i} }{\partial x_{j} } \frac{\partial u_{<del class="diffchange diffchange-inline">i</del>} }{\partial x_{<del class="diffchange diffchange-inline">j</del>} } \right\rangle <del class="diffchange diffchange-inline">+ </del>\left\langle \frac{\partial u_{i} }{\partial x_{j} } \frac{\partial u_{j} }{\partial x_{i} } \right\rangle \right\}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\epsilon = 2\nu \left\langle s_{ij} s_{ij} \right\rangle = \nu \left\{ \left\langle \frac{\partial u_{i} }{\partial x_{j} } <ins class="diffchange diffchange-inline">+ </ins>\frac{\partial u_{<ins class="diffchange diffchange-inline">j</ins>} }{\partial x_{<ins class="diffchange diffchange-inline">i</ins>} } \right\rangle \left\langle \frac{\partial u_{i} }{\partial x_{j} } <ins class="diffchange diffchange-inline">+ </ins>\frac{\partial u_{j} }{\partial x_{i} } \right\rangle \right\}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(14)</td></tr></table></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></td><td width="5%">(14)</td></tr></table></div></td></tr>
</table>Malkavian GThttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=9999&oldid=prevMech E: /* Kinetic energy of the mean motion and production of turbulence */2010-02-24T02:11:04Z<p><span class="autocomment">Kinetic energy of the mean motion and production of turbulence</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Unlike the fluctuating equations, there is no need to average here, since all the terms are already averages.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Unlike the fluctuating equations, there is no need to average here, since all the terms are already averages.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In exactly the same manner that we <del class="diffchange diffchange-inline">rearrannged </del>the terms in the <del class="diffchange diffchange-inline">eqyation </del>for the kinetic energy of the fluctuations, we can rearrange the equation for the kinetic energy of the mean flow to obtain:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In exactly the same manner that we <ins class="diffchange diffchange-inline">rearranged </ins>the terms in the <ins class="diffchange diffchange-inline">equation </ins>for the kinetic energy of the fluctuations, we can rearrange the equation for the kinetic energy of the mean flow to obtain:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><table width="70%"><tr><td></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><table width="70%"><tr><td></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The role of all of the terms can immediately be recognized since each term has its counterpart in the equation for the average fluctuating kinetic energy. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The role of all of the terms can immediately be recognized since each term has its counterpart in the equation for the average fluctuating kinetic energy. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Comparison of equations 23 and 6 reveals that the term <math>-\left\langle u_{i}u_{j}\right\rangle \partial U_{i}/\partial x_{j}</math> appears in the equations for the <del class="diffchange diffchange-inline">kinetis </del>energy of BOTH the mean and the fluctuations. There is, however, one VERY important difference. This "production" term has the opposite sign in the <del class="diffchange diffchange-inline">equationfor </del>the mean kinetic energy than in that for the mean fluctuating kinetic energy! Therefore, ''whatever its effect on the kinetic energy of the mean, its effect on the kinetic energy of the fluctuations will be the opposite''. Thus kinetic energy can be interchanged between the mean and fluctuating motions. In fact, the only other term involving fluctuations in the equation for the kinetic energy of the mean motion is divergence term; therefore it can only move the kinetic energy of the mean flow from one place to another. Therefore this "production" term provides the ''only'' means by which energy can be interchanged between the mean flow and fluctuations.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Comparison of equations 23 and 6 reveals that the term <math>-\left\langle u_{i}u_{j}\right\rangle \partial U_{i}/\partial x_{j}</math> appears in the equations for the <ins class="diffchange diffchange-inline">kinetic </ins>energy of BOTH the mean and the fluctuations. There is, however, one VERY important difference. This "production" term has the opposite sign in the <ins class="diffchange diffchange-inline">equation for </ins>the mean kinetic energy than in that for the mean fluctuating kinetic energy! Therefore, ''whatever its effect on the kinetic energy of the mean, its effect on the kinetic energy of the fluctuations will be the opposite''. Thus kinetic energy can be interchanged between the mean and fluctuating motions. In fact, the only other term involving fluctuations in the equation for the kinetic energy of the mean motion is divergence term; therefore it can only move the kinetic energy of the mean flow from one place to another. Therefore this "production" term provides the ''only'' means by which energy can be interchanged between the mean flow and fluctuations.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Understanding the manner in which this energy exchange between mean and fluctuating motions is accomplished represents one of the most challenging problems in turbulence. The overall exchange can be understood by exploiting the analogy which treats <math>-\rho \left\langle u_{i}u_{j}\right\rangle </math> as a stress, the Reynolds stress. The term <math>-\rho \left\langle u_{i}u_{j}\right\rangle \partial U_{i}/\partial x_{j}</math> can be thought of as the working of the Reynolds stress against the mean velocity gradient of the flow, exactly as the viscous stresses resist deformation by the instantaneous velocity gradients. This energy expended against the Reynolds stress during deformation by the mean motion ends up in the fluctuating motions, however, while that expended against viscous stresses goes directly to internal energy. As we have already seen, the viscous deformation work from the fluctuating motions (or dissipation) will eventually send this fluctuating kinetic energy on to internal energy as well.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Understanding the manner in which this energy exchange between mean and fluctuating motions is accomplished represents one of the most challenging problems in turbulence. The overall exchange can be understood by exploiting the analogy which treats <math>-\rho \left\langle u_{i}u_{j}\right\rangle </math> as a stress, the Reynolds stress. The term <math>-\rho \left\langle u_{i}u_{j}\right\rangle \partial U_{i}/\partial x_{j}</math> can be thought of as the working of the Reynolds stress against the mean velocity gradient of the flow, exactly as the viscous stresses resist deformation by the instantaneous velocity gradients. This energy expended against the Reynolds stress during deformation by the mean motion ends up in the fluctuating motions, however, while that expended against viscous stresses goes directly to internal energy. As we have already seen, the viscous deformation work from the fluctuating motions (or dissipation) will eventually send this fluctuating kinetic energy on to internal energy as well.</div></td></tr>
</table>Mech Ehttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=8966&oldid=prevJola at 09:19, 25 February 20082008-02-25T09:19:02Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{Turbulence credit wkgeorge}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{Turbulence credit wkgeorge}}</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>{{Chapter navigation|Reynolds averaged equations|Stationarity and <del class="diffchange diffchange-inline">homogenity</del>}}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>{{Chapter navigation|Reynolds averaged equations|Stationarity and <ins class="diffchange diffchange-inline">homogeneity</ins>}}</div></td></tr>
</table>Jolahttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=8702&oldid=prevMichail: finished :-))2007-12-26T08:43:20Z<p>finished :-))</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Models accounting for this are said to include a "''return-to-isotropy''" term. An additional term must also be included to account for the direct effect of the mean shear on the pressure-strain rate correlation, and this is reffered to as the "''rapid term''". The reasons for this latter term are not easy to see from single point equations, but fall out rather naturally from the two-point Reynolds stress equations we shall discuss later.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Models accounting for this are said to include a "''return-to-isotropy''" term. An additional term must also be included to account for the direct effect of the mean shear on the pressure-strain rate correlation, and this is reffered to as the "''rapid term''". The reasons for this latter term are not easy to see from single point equations, but fall out rather naturally from the two-point Reynolds stress equations we shall discuss later.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">== * * ==</del></div></td><td colspan="2"> </td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{Turbulence credit wkgeorge}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{Turbulence credit wkgeorge}}</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{Chapter navigation|Reynolds averaged equations|Stationarity and homogenity}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{Chapter navigation|Reynolds averaged equations|Stationarity and homogenity}}</div></td></tr>
</table>Michailhttps://cfd-online.com/W/index.php?title=Introduction_to_turbulence/Turbulence_kinetic_energy&diff=8701&oldid=prevMichail: /* The Intercomponent Transfer of Energy */2007-12-26T08:42:37Z<p><span class="autocomment">The Intercomponent Transfer of Energy</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Example:''' In simple turbulent free shear flows like wakes or jets where the energy is primarily produced in a single component (as in the example above), typically <math> \left\langle u^{2}_{1} \right\rangle \approx \left\langle u^{2}_{2} \right\rangle + \left\langle u^{2}_{3} \right\rangle </math> where <math> \left\langle u^{2}_{1} \right\rangle </math> is the kinetic of the component produced directly by the action of Reynolds stresses against the mean velocity gradient. Moreover <math> \left\langle u^{2}_{2} \right\rangle \approx \left\langle u^{2}_{3} \right\rangle </math>. This, of course, makes some sense in light of the above, since both off-axis components get most of their energy from the pressure-strain rate terms.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Example:''' In simple turbulent free shear flows like wakes or jets where the energy is primarily produced in a single component (as in the example above), typically <math> \left\langle u^{2}_{1} \right\rangle \approx \left\langle u^{2}_{2} \right\rangle + \left\langle u^{2}_{3} \right\rangle </math> where <math> \left\langle u^{2}_{1} \right\rangle </math> is the kinetic of the component produced directly by the action of Reynolds stresses against the mean velocity gradient. Moreover <math> \left\langle u^{2}_{2} \right\rangle \approx \left\langle u^{2}_{3} \right\rangle </math>. This, of course, makes some sense in light of the above, since both off-axis components get most of their energy from the pressure-strain rate terms.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>It is possible to show that the pressure-strain rate terms vanish in isotropic turbulence. This suggests (at least to some) that the natural state for turbulence in the absence of other influences is the <del class="diffchange diffchange-inline">9isotropic </del>state. This has also been exploited by the turbulence modelers. One of the most common assumptions involves setting these pressure-strain rate terms (as they occur in the Reynolds shear equation) proportional to the anisotropy of the flow defined by:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>It is possible to show that the pressure-strain rate terms vanish in isotropic turbulence. This suggests (at least to some) that the natural state for turbulence in the absence of other influences is the <ins class="diffchange diffchange-inline">isotropic </ins>state. This has also been exploited by the turbulence modelers. One of the most common assumptions involves setting these pressure-strain rate terms (as they occur in the Reynolds shear equation) proportional to the anisotropy of the flow defined by:</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">a_{ij} = \left\langle u_{i} u_{j} \right\rangle - \left\langle q^{2} \right\rangle \delta_{ij} / 3</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"></math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"></td><td width="5%">(46)</td></tr></table></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Models accounting for this are said to include a "''return-to-isotropy''" term. An additional term must also be included to account for the direct effect of the mean shear on the pressure-strain rate correlation, and this is reffered to as the "''rapid term''". The reasons for this latter term are not easy to see from single point equations, but fall out rather naturally from the two-point Reynolds stress equations we shall discuss later.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== * * ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== * * ==</div></td></tr>
</table>Michail