Draft:Boundary Layer Separation in Forced-Natural Convection

• Comment: I've asked for comments on this draft at Talk:Flow separation. Curb Safe Charmer (talk) 13:47, 23 September 2024 (UTC)

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Boundary layer separation is a critical phenomenon in fluid dynamics that occurs when the flow of fluid detaches from the surface of an object. In combined forced convection and natural convection systems, this process is even more complex, involving both external forces driving the flow and internal buoyancy forces due to temperature gradients.

Boundary layer separation in combined convection occurs when the flow detaches from the surface due to the inability of the boundary layer to withstand the effects of both inertial and buoyancy forces. This detachment reduces heat transfer efficiency, increases drag, and can result in flow recirculation. It is critical in the design of heat exchangers., aerodynamics , and cooling systems for electronics .

Governing equations

Boundary layer theory is based on simplifying the Navier-Stokes equations under the assumption that viscous effects are confined to a thin layer near the surface . For combined forced and natural convection, the momentum and energy equations take the form:

Continuity equation:

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$

Momentum equation (x-direction):

$\rho \left({\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}\right)=-{\frac {\partial p}{\partial x}}+\mu {\frac {\partial ^{2}u}{\partial y^{2}}}+\rho g\beta (T-T_{\infty })$

Here, the term $\rho g\beta (T-T_{\infty })$ accounts for the buoyancy force caused by temperature gradients. It couples natural convection with the flow driven by the external force .

Momentum equation (y-direction):

${\frac {\partial p}{\partial y}}=0$

In boundary layer theory, the pressure gradient in the y-direction is assumed to be negligible due to the thinness of the boundary layer. However, the x-direction momentum equation shows that the adverse pressure gradient may cause flow separation if the wall shear stress becomes zero .

Energy equation:

$\rho c_{p}\left({\frac {\partial T}{\partial t}}+u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}\right)=k{\frac {\partial ^{2}T}{\partial y^{2}}}$

This equation governs the temperature distribution, where:

• T is the temperature,
• c_p is the specific heat capacity,
• k is the thermal conductivity.

Boundary layer development

The boundary layer is a thin region near the surface where the velocity and temperature gradients are significant. The development of this layer is critical in understanding when separation occurs. For combined convection, both forced and natural convection influence the boundary layer growth .

Velocity boundary layer

In forced convection, the boundary layer develops as the fluid adheres to the surface, and viscous forces cause a velocity gradient from the wall to the free stream. The Blasius solution for a flat plate gives the velocity profile for laminar flow in the absence of buoyancy :

$u(x,y)=U_{\infty }f'(\eta )\quad {\text{where}}\quad \eta ={\frac {y}{\sqrt {\frac {\nu x}{U_{\infty }}}}}$

where $f'(\eta )$ is the non-dimensional velocity profile, $U_{\infty }$ is the free stream velocity, and $\nu$ is the kinematic viscosity.

When buoyancy is included, the momentum equation has an additional source term proportional to the temperature difference . The velocity boundary layer grows more quickly due to the influence of the buoyant force, but it is also more prone to separation .

Thermal boundary layer

The thermal boundary layer describes the region where temperature gradients are significant. Its thickness is influenced by both conduction and convection processes. For laminar flow, the energy equation for the thermal boundary layer leads to the following approximate solution :

$T(x,y)=T_{w}+(T_{\infty }-T_{w})\left(1-{\text{erf}}\left({\frac {y}{\sqrt {\frac {2kx}{\rho c_{p}U_{\infty }}}}}\right)\right)$

where:

• $T_{w}$ is the wall temperature,
• $T_{\infty }$ is the free stream temperature.

The ratio of the thermal to velocity boundary layer thickness is characterized by the Prandtl number:

$Pr={\frac {\mu c_{p}}{k}}$

If $Pr\gg 1$, the thermal boundary layer is thinner than the velocity boundary layer, which is typical for liquids. If $Pr\ll 1$, the thermal boundary layer is thicker, common in gases

Boundary layer separation

Boundary layer separation occurs when the velocity gradient at the wall becomes zero or negative, resulting in the flow detaching from the surface . The criteria for boundary layer separation can be derived from the momentum equation. Using the Bernoulli equation and boundary layer approximation, we can express the condition for separation as:

${\frac {\partial p}{\partial x}}>0\quad {\text{(adverse pressure gradient)}}$

Separation happens when the adverse pressure gradient is large enough to reverse the flow near the wall, causing a stagnation point:. In combined convection, this phenomenon is influenced by both inertial and buoyancy forces.

To predict separation, we analyze the local Reynolds number (Re) and Grashof number (Gr):

$Re_{x}={\frac {U_{\infty }x}{\nu }}\quad {\text{and}}\quad Gr_{x}={\frac {g\beta (T_{w}-T_{\infty })x^{3}}{\nu ^{2}}}$

The interaction between Re and Gr leads to the Richardson number (Ri), which indicates the relative importance of natural to forced convection :

$Ri={\frac {Gr_{x}}{Re_{x}^{2}}}$

When $Ri\gg 1$, natural convection dominates, while $Ri\ll 1$ means forced convection is dominant. The critical value of Ri determines when boundary layer separation is likely to occur.

Critical parameters

Several dimensionless numbers are critical in determining whether boundary layer separation occurs in combined convection:

• The Reynolds number (Re), which characterizes the inertial forces relative to viscous forces :

$Re_{x}={\frac {U_{\infty }x}{\nu }}$

• The Grashof number (Gr), which characterizes the buoyancy forces relative to viscous forces :

$Gr_{x}={\frac {g\beta (T_{w}-T_{\infty })x^{3}}{\nu ^{2}}}$

• The Richardson number (Ri), which compares the effects of natural to forced convection

$Ri={\frac {Gr_{x}}{Re_{x}^{2}}}$

These parameters are essential for designing thermal systems where the balance between forced and natural convection affects the flow behavior.

Engineering applications

Boundary layer separation in combined convection is relevant in several engineering disciplines:

• In heat exchangers, separation decreases heat transfer efficiency and increases pressure drop/
• In aerodynamics, separation increases drag, leading to flow instability and energy loss.
• In electronics cooling, separation decreases the effectiveness of heat sinks, leading to increased temperatures in critical components.