Editing Boundary layer (section)

==Turbulent boundary layers==

The treatment of turbulent boundary layers is far more difficult due to the time-dependent variation of the flow properties.  One of the most widely used techniques in which turbulent flows are tackled is to apply [[Reynolds decomposition]]. Here the instantaneous flow properties are decomposed into a mean and fluctuating component with the assumption that the mean of the fluctuating component is always zero. Applying this technique to the boundary layer equations gives the full turbulent boundary layer equations not often given in literature:

:<math> {\partial \overline{u}\over\partial x}+{\partial \overline{v}\over\partial y}=0 </math>

:<math> \overline{u}{\partial \overline{u} \over \partial x}+\overline{v}{\partial \overline{u} \over \partial y}=-{1\over \rho} {\partial \overline{p} \over \partial x}+ \nu \left({\partial^2 \overline{u}\over \partial x^2}+{\partial^2 \overline{u}\over \partial y^2}\right)-\frac{\partial}{\partial y}(\overline{u'v'})-\frac{\partial}{\partial x}(\overline{u'^2}) </math>

:<math> \overline{u}{\partial \overline{v} \over \partial x}+\overline{v}{\partial \overline{v} \over \partial y}=-{1\over \rho} {\partial \overline{p} \over \partial y}+\nu \left({\partial^2 \overline{v}\over \partial x^2}+{\partial^2 \overline{v}\over \partial y^2}\right)-\frac{\partial}{\partial x}(\overline{u'v'})-\frac{\partial}{\partial y}(\overline{v'^2}) </math>

Using a similar order-of-magnitude analysis, the above equations can be reduced to leading order terms. By choosing length scales <math> \delta </math> for changes in the transverse-direction, and <math> L </math> for changes in the streamwise-direction, with <math>\delta<<L</math>, the x-momentum equation simplifies to:

:<math> \overline{u}{\partial \overline{u} \over \partial x}+\overline{v}{\partial \overline{u} \over \partial y}=-{1\over \rho} {\partial \overline{p} \over \partial x}+{\nu}{\partial^2 \overline{u}\over \partial y^2}-\frac{\partial}{\partial y}(\overline{u'v'}). </math>

This equation does not satisfy the [[no-slip condition]] at the wall. Like Prandtl did for his boundary layer equations, a new, smaller length scale must be used to allow the viscous term to become leading order in the momentum equation. By choosing <math>\eta<<\delta</math> as the ''y''-scale, the leading order momentum equation for this "inner boundary layer" is given by:

:<math> 0=-{1\over \rho} {\partial \overline{p} \over \partial x}+{\nu}{\partial^2 \overline{u}\over \partial y^2}-\frac{\partial}{\partial y}(\overline{u'v'}). </math>

In the limit of infinite Reynolds number, the pressure gradient term can be shown to have no effect on the inner region of the turbulent boundary layer. The new "inner length scale" <math>\eta</math> is a viscous length scale, and is of order <math>\frac{\nu}{u_*}</math>, with <math>u_*</math> being the velocity scale of the turbulent fluctuations, in this case a [[Shear velocity#Friction Velocity in Turbulence|friction velocity]].

Unlike the laminar boundary layer equations, the presence of two regimes governed by different sets of flow scales (i.e. the inner and outer scaling) has made finding a universal similarity solution for the turbulent boundary layer difficult and controversial. To find a similarity solution that spans both regions of the flow, it is necessary to asymptotically match the solutions from both regions of the flow. Such analysis will yield either the so-called [[Law of the wall|log-law]] or [[Law of the wall#Power law solutions|power-law]].

Similar approaches to the above analysis has also been applied for thermal boundary layers, using the energy equation in compressible flows.<ref>{{Cite journal |last=von Karman |first=T. |date=1939 |title=The analogy between fluid friction and heat transfer. |journal=Transactions of the American Society of Mechanical Engineers |volume=61 |issue=8 |pages=705–710|doi=10.1115/1.4021298 |s2cid=256805665 }}</ref><ref>{{Cite journal |last1=Guo |first1=J. |last2=Yang |first2=X. I. A. |last3=Ihme |first3=M. |date=March 2022 |title=Structure of the thermal boundary layer in turbulent channel flows at transcritical conditions |journal=Journal of Fluid Mechanics |language=en |volume=934 |doi=10.1017/jfm.2021.1157 |bibcode=2022JFM...934A..45G |s2cid=246066677 |issn=0022-1120|doi-access=free }}</ref>

The additional term <math>\overline{u'v'}</math> in the turbulent boundary layer equations is known as the Reynolds shear stress and is unknown [[A priori and a posteriori|a priori]]. The solution of the turbulent boundary layer equations therefore necessitates the use of a [[Turbulence modeling|turbulence model]], which aims to express the Reynolds shear stress in terms of known flow variables or derivatives. The lack of accuracy and generality of such models is a major obstacle in the successful prediction of turbulent flow properties in modern fluid dynamics.

A constant stress layer exists in the near wall region. Due to the damping of the vertical velocity fluctuations near the wall, the Reynolds stress term will become negligible and we find that a linear velocity profile exists. This is only true for the very [[Law of the wall#Near the wall|near wall region]].