Fluid Dynamics Basics

Origins of Normal Forces, Shear Forces and Boundary Layer

Fluid molecules move randomly, carrying their mass, momentum and energy from one place to another.

When the fluid is static, there is no fluid flow. Fluid molecules collide without sliding at the contact faces. Normal forces (= pressure forces) exist.
When the fluid is in motion, due to the velocity differences within the fluid, molecules also collide with sliding at the contact faces. Shear forces (= tangential forces, friction forces) arise from the internal friction of the fluid.

The viscosity \(\mu\) is a measure of the internal frictional resistance of a fluid to flow motion.

Consider a flow around a solid body. Due to viscosity, there is a velocity profile of flow, zero velocity at the solid surface and increasing to freestream velocity as you move away from the surface. The boundary layer is this thin layer of flow near the surface of solid bodies. Outside of the boundary layer, the flow is essentially frictionless and can be approximated as inviscid. The boundary layer can be either laminar or turbulent.

Flow separations are interactions between the boundary layer and freestream, that arise from the stall of flow.

Ideal Fluid and Real Fluid

Inviscid fluids (= ideal fluids) have no viscosity and, there are no frictional shear forces even if the fluid is in motion.
Real fluids generate frictional shear forces when the fluids are moving.

Incompressible Fluid

An incompressible fluid has a constant density \(\rho\). The density of air is affected by changes in temperature and pressure.

Real gases including air follow the ideal gas law approximately \[ \begin{array}{l} \begin{align} pV &= mRT \\ p &= \rho RT \\ \end{align}\end{array} \]

where
\(p\) is the absolute pressure in Pa
\(V\) volume in m^3
\(m\) mass in kg
\(R\) gas constant in J/(kg⋅K)
\(T\) absolute temperature in K
\(\rho\) density in kg/m^3

Laminar and Turbulent Flow

Laminar and turbulent are the two types of fluid flow.
In laminar flow, the fluid flows smoothly in layers without interchanges between the layers.
In turbulent flow, chaotic thorough mixing of fluid occurs.

Reynolds experiment consists of passing fluid inside a pipe of circular diameter with a thin stream of dye injected into the flow. The dye is for visualising the flow scheme. The experiment observed that the type of flow is affected by the pipe diameter, flow speed and viscosity of the fluid.

A dimensionless parameter, the Reynolds number is \[\text{Re} = \dfrac{\rho l u}{\mu} = \dfrac{l u}{\nu} \] where
\(\rho\) is the density of fluid in kg/m^3
\(l\) characteristic length of the body, the pipe diameter in m
\(u\) mean axial flow in m/s
\(\mu\) viscosity of the fluid in Pa s
\(\nu\) kinematic viscosity in m^2 /s, \(\nu = \dfrac{\mu}{\rho}\)

For circular pipe flow, the flow turns to turbulent at approximately \(\text{Re}>2100\).

Bernoulli’s Equation

The Bernoulli’s equation is applicable to any two points in the flow of an incompressible inviscid fluid, i.e. the outside the boundary layer.

The equation represents the conversion of pressure and velocity when the fluid flows smoothly around a body. The surface pressure on the body and lift force can be well predicted, particularly when the boundary layer is thin and no flow separation occurs. (The drug force arises from shear forces and cannot be predicted by the Bernoulli’s equation.)

The Bernoulli’s equation \[ p + \dfrac{1}{2} \rho v^2 + \rho g z = p_0 = \text{constant} \quad \text{[Pa]} \] where
\(v \quad\) flow velocity in m/s
\(p \quad\) static pressure in Pa. This pressure can be measured by a manometer pressure gauge.
\(\dfrac{1}{2} \rho v^2 \quad \) dynamic pressure
\( \rho g z \quad \) hydrostatic pressure
\(p_0 \quad\) total pressure
\(p + \dfrac{1}{2} \rho v^2 \quad \) stagnation pressure

Divide by the density \( \rho \) \[ \dfrac{p}{\rho} + \dfrac{1}{2}v^2 + gz = \text{constant} \quad \text{[J/kg]} = \text{[Nm/kg]} \] where
\(\dfrac{p}{\rho} \quad \) pressure energy. The energy per unit mass required to maintain the volume of fluid against the external pressure.
\( \dfrac{1}{2}v^2 \quad\) kinetic energy. The amount of work per unit mass required to accelerate the unit mass of fluid to the velocity \(v\).
\( gz \quad\) potential energy. Related to the height change of the fluid. Negligible for flow around automotive vehicle.

Pressure Distribution around Aerofoils

Bernoulli’s equation implies that when the dynamic pressure is increased, the static pressure must decrease, and vice versa.
When the air flows around an aerofoil, the air accelerates and pass over the top and bottom surfaces. The top flow accelerates to a quicker velocity than the bottom frow, since the curvature rate is higher at the top and the flow need to travel greater distance.

Since the pressure over the top of the aerofoil is lower than that of the bottom, the aerofoil generates a lift force. The below figure illustrates the reduction in static pressure \(\Delta p \), relative to the free stream. Ideal fluid is assumed, and the pressure arrows were drawn perpendicular to the surfaces.

The sum of the pressure components perpendicular to the free stream is the lift force. The sum of the tangential components equal to zero, due to a complete recovery of static pressure with the assumption of the ideal fluid.

References

[1] W. Milliken and D. Milliken, Race Car Vehicle Dynamics.
[2] C. Crowe et al., Engineering Fluid Mechanics.