Fundamental Fluid Dynamics Theory

Last edited: 2024-10-28 11:56:03

In fluid dynamics, there are many concepts, formulas, and coefficients. In this post the most fundamental ones are explained

Bernoulli's Principle

In incompressible isentropic flow, i.e. a flow where a negligible amount of energy is lost to non-reversible processes such as heat in turbulent flows, one can derive Bernoulli's principle from the conservation of energy

$$\frac{u^2}{2}+\frac{p}{\rho}+z\cdot g= \text{constant},$$

where $z$ is the vertical position of a fluid element, this law states that an increase in velocity implies a decrease in pressure and vice versa.

Dimensionless parameters

Dimensionless parameters are used to make a parameter comparable with parameters of other types of flow. Below the relevant parameters are presented.

Reynolds Number Formula

The Reynolds number, Re, is a dimensionless number that tells the relation between the inertia and viscosity in a newtonian fluid and is defined by the following equation

$$\text{Re} = \frac{\rho V L}{\mu} = \left\{ \frac{\text{Inertia}}{\text{Viscosity}} \right\}.$$

$V$ and $L$ are the characteristic velocity and length of the flow. A high Re is associated with a fast, large-scale turbulent flow, whereas a low Re corresponds to a slow, viscous flow. Fast flows of gas imply a relatively high Re.

Drag Coefficient Formula

The drag coefficient of a body, $C_D$, is a dimensionless number indicative of drag force. $C_D$ is defined by the following equation

$$C_D = \frac{F_D}{\frac{1}{2}\rho u_{\infty}^2 A},$$

where $A$ is the projected area from the front of the body and the denominator $(\frac{1}{2} \rho u_{\infty}^{2})$ is the dynamic pressure of the free stream.

Lift Coefficient Formula

The lift coefficient of a body, $C_L$ is defined by the following equation

$$C_L = \frac{F_L}{\frac{1}{2}\rho u_{\infty}^2 A},$$

where $A$ is the projected area from the front of the body. $C_L$ is a dimensionless coefficient that increases with lift force. Depending on the area of study, $C_L$ will have different directions. For example, when designing race cars, $C_L$ will indicate the lift in the negative $z$ direction, i.e., a high downforce will correspond to a large $C_L$. When designing airplanes it will be in the other direction to indicate lift.

Pressure Coefficient Formula

The pressure coefficient, $C_p$, is a dimensionless parameter that describes the relative pressure. $C_p$ is defined as

$$C_{p}=\frac{p-p_{\infty}}{\frac{1}{2} \rho u_{\infty}^{2}},$$

where $p$ is the static pressure at the point where $C_p$ is calculated, while $p_{\infty}$ is the static pressure in the free stream.

Skin Friction Coefficient Formula

The skin friction coefficient, $C_f$, is a dimensionless parameter defined as

$$C_{f}=\frac{\tau _{w}}{\frac{1}{2} \rho u_{\infty}^{2}}.$$

Vorticity Formula

The vorticity, $\vec{\zeta}$, is defined as the curl of the velocity field, $\nabla\,\times\,\vec{u}$, which relates to the rotation of a velocity field. If the vorticity is zero the flow has no rotation and is called irrotational. The vorticity in the $x$-direction is given by

$$\zeta_x= \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}.$$

Navier-Stokes Equations

A fluid can be described with the Navier-Stokes equations if it is incompressible and newtonian as

$$\rho \left( \frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla) \vec{u} \right) = \rho \vec{g} - \nabla p + \mu \Delta \vec{u}.$$

RANS Formula

In turbulent flow, some fluctuations cause rapid changes in the velocity and pressure in the Navier-stokes equations. These rapid changes can be easier to account for if the velocity and pressure are split into a mean variable $\bar{u}$ and a fluctuation variable $u'$. The velocity in the $x$-direction is therefore $u = \bar{u} + u'$. If those splits are put into the Navier-stokes equations we get the Reynold's Averaged Navier Stokes (RANS) equations after some derivations. The RANS in $x$-direction is

$$\rho\frac{d \overline{u}}{dt}=-\frac{\partial \overline{p}}{\partial x}+\rho g_x+\frac{\partial }{\partial x}\left(\mu\frac{\partial \overline{u}}{\partial x}-{\rho \overline{u^{\prime 2}}}\right)$$ $$+\frac{\partial }{\partial y}\left(\mu\frac{\partial \overline{u}}{\partial y}-{\rho \overline{u^{\prime}v^{\prime}}}\right)+\frac{\partial }{\partial z}\left(\mu\frac{\partial \overline{u}}{\partial z}-{\rho \overline{u^{\prime}w^{\prime}}}\right).$$

Flow Past Boundary

Bodies, such as plates, immersed in a fluid stream create a boundary layer flow. The boundary layer is defined as the region where the flow velocity is less than 99 percent of the velocity of the external flow ($u_\infty$). The boundary layer flow is made up of two parts, the laminar part and the turbulent part:

$$\frac{\delta}{x} \approx \left\{ \begin{aligned} & \frac{5}{\text{Re}_x^{1/2}} \qquad 10^3 < \text{Re}_x < 10^6 \text{ laminar flow} \\ & \frac{0.16}{\text{Re}_x^{1/7}} \qquad 10^6 < \text{Re}_x \text{ turbulent flow} \end{aligned} \right. .$$

$\delta$ is the thickness of the boundary layer and $\text{Re}_x$ the Reynolds number at $x$. An illustration of he boundary level flow can be seen in figure below.

Flow Separation

Backflow, i.e. flow going in the opposite direction of the freestream, at the wall indicates that the flow is separated. The point of separation is when the wall shear is 0 which means the velocity gradient is 0.

The Logarithmic Overlap Law

The turbulent flow near a wall can be broken up into three regions based on which shear stress dominates:

• Wall layer: Viscous shear dominates
• Outer layer: Turbulent shear dominates
• Overlap layer: A mix of the shear types

The inner layers velocity can be modeled as $u = f(\mu, \tau_w, \rho, y)$, $\tau_w$ is the wall shear stress. By dimensional analysis the following relation can be deducted:

$$u^+ = \frac{u}{u^*} = F \left(\frac{y u^*}{\nu} \right),$$

where $u^* = \sqrt{\tau_w/\rho}$. An akin method can be done with the outer layer, $(u_\infty - u)_{\text{outer}} = g(\delta, \tau_w, \rho, y)$ and the dimensional analysis leads to:

$$\frac{u_\infty - u}{u^*} = G\left( \frac{y}{\delta} \right).$$

The overlap layer makes it so these two layers overlap smoothly and for this to be true the overlap velocity must vary logarithmically with y:

$$\frac{u}{u^*} = \frac{1}{\kappa} \ln{\frac{y u^*}{\nu}} + B,$$

where $\kappa \approx 0.41$ and $B \approx 5.0$. We define the dimensionless number $y^+$ as

$$y^+ =\frac{y u^*}{\nu},$$

where $u^* = \sqrt{\tau_w/\rho}$. For $y^+ \in [0,5]$ the inner layer dominates and is proportional to $y^+$. It can be described with the dimensionless variable

$$u^+ = \frac{u}{u^*} = y^+.$$

For $y^+ \in [30, 10^3]$ the overlap layers dominates and can be described as

$$u^+ = \frac{1}{\kappa} \ln{y^+} + B,$$

where $\kappa \approx 0.41$ and $B \approx 5.0$. For $y^+ \in [5,30]$ the inner layer has to curve so it merges with the overlap layer. Only experimental data exists for this interval making it difficult to model.