How to find velocity profile in terms of boundary layer thickness?
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The boundary layer thickness, δ, is the distance across a boundary layer from the walls to a point where the flow velocity has essentially reached the 'free stream' velocity, {\displaystyle u_{0}}. This distance is defined normal to the wall. It is customarily defined as the point {\displaystyle y_{99}} where:
{\displaystyle u(y_{99})=0.99u_{o}}
at a point on the wall {\displaystyle x}. For laminar boundary layers over a flat plate, the Blasius solution to the flow governing equations gives:[2]
{\displaystyle \delta \approx 4.91{\sqrt {{\nu x} \over u_{0}}}}{\displaystyle \delta \approx 4.91x/{\sqrt {\mathrm {Re} _{x}}}}
For turbulent boundary layers over a flat plate, the boundary layer thickness is given by:[3]
{\displaystyle \delta \approx 0.37x/{\mathrm {Re} _{x}}^{1/5}}
where
{\displaystyle \mathrm {Re} _{x}=\rho u_{0}x/\mu }{\displaystyle \delta } is the overall thickness (or height) of the boundary layer{\displaystyle \mathrm {Re} _{x}} is the Reynolds number{\displaystyle \rho } is the density{\displaystyle u_{0}} is the freestream velocity{\displaystyle x} is the distance downstream from the start of the boundary layer{\displaystyle \mu } is the dynamic viscosity{\displaystyle \nu =\mu /\rho } is the kinematic viscosity
The turbulent boundary layer thickness formula assumes 1) the flow is turbulent right from the start of the boundary layer and 2) the turbulent boundary layer behaves in a geometrically similar manner (i.e. the velocity profiles are geometrically similar along the flow in the x-direction, differing only by stretching factors in {\displaystyle y} and {\displaystyle u(y)}[4]). Neither one of these assumptions are true for the general turbulent boundary layer case so care must be excersised in applying this formula.
The velocity thickness can also be referred to as the Soole ratio, although the gradient of the thickness over distance would be adversely proportional to that of velocity thickness.
{\displaystyle u(y_{99})=0.99u_{o}}
at a point on the wall {\displaystyle x}. For laminar boundary layers over a flat plate, the Blasius solution to the flow governing equations gives:[2]
{\displaystyle \delta \approx 4.91{\sqrt {{\nu x} \over u_{0}}}}{\displaystyle \delta \approx 4.91x/{\sqrt {\mathrm {Re} _{x}}}}
For turbulent boundary layers over a flat plate, the boundary layer thickness is given by:[3]
{\displaystyle \delta \approx 0.37x/{\mathrm {Re} _{x}}^{1/5}}
where
{\displaystyle \mathrm {Re} _{x}=\rho u_{0}x/\mu }{\displaystyle \delta } is the overall thickness (or height) of the boundary layer{\displaystyle \mathrm {Re} _{x}} is the Reynolds number{\displaystyle \rho } is the density{\displaystyle u_{0}} is the freestream velocity{\displaystyle x} is the distance downstream from the start of the boundary layer{\displaystyle \mu } is the dynamic viscosity{\displaystyle \nu =\mu /\rho } is the kinematic viscosity
The turbulent boundary layer thickness formula assumes 1) the flow is turbulent right from the start of the boundary layer and 2) the turbulent boundary layer behaves in a geometrically similar manner (i.e. the velocity profiles are geometrically similar along the flow in the x-direction, differing only by stretching factors in {\displaystyle y} and {\displaystyle u(y)}[4]). Neither one of these assumptions are true for the general turbulent boundary layer case so care must be excersised in applying this formula.
The velocity thickness can also be referred to as the Soole ratio, although the gradient of the thickness over distance would be adversely proportional to that of velocity thickness.
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