CFD Applications

Method

Darcy-Weisbach: $\Delta p=f\dfrac LD\dfrac{\rho V^2}2$, with $f$ from the Colebrook equation

$$\frac1{\sqrt f}=-2\log_{10}\left(\frac{\varepsilon/D}{3.7}+\frac{2.51}{Re\sqrt f}\right)$$

For parallel pipes: the pressure drop is the same, and $Q_B+Q_C=Q$. Iterate on the split.

Pipe A (series, full flow)

$V=\dfrac{4Q}{\pi D^2}=\dfrac{4\times0.03}{\pi\times0.15^2}=1.698$ m/s; $Re=\dfrac{998\times1.698\times0.15}{10^{-3}}=2.54\times10^5$; $\varepsilon/D=3.07\times10^{-4}$.

Colebrook gives $f=0.0173$.

$$\Delta p_A=0.0173\times\frac{300}{0.15}\times\frac{998\times1.698^2}{2}=4.98\times10^4\ \text{Pa}$$

Parallel pipes B and C

Trial splits are adjusted until $\Delta p_B=\Delta p_C$. The converged split is

$Q_B+Q_C=0.0300$ m³/s, as required. Pipe B (larger) carries 64% of the flow.

Total

$$\Delta p_{total}=\Delta p_A+\Delta p_{parallel}=49.8+110.1=159.9\ \text{kPa}$$

(about 16.3 m of water head).

Use of CFD for pipe networks

• A full 3D CFD model of a part (bends, tees, valves, manifolds) gives loss coefficients $K$ and velocity maps, including flow maldistribution at junctions.
• Typical set-up: velocity inlet, pressure outlet, no-slip walls with roughness, RANS k-$\varepsilon$ or k-$\omega$ SST, prism layers on walls; pressure drop = difference of area-averaged total pressure.
• For large networks, 1D network solvers (using $f$ and $K$ from CFD or correlations) are coupled with 3D CFD at critical components.
• Results are checked against Darcy-Weisbach and experiments.

Answer: $Q_B=0.0193$ m³/s, $Q_C=0.0107$ m³/s; total $\Delta p\approx160$ kPa.

(a) Reynolds number

$$Re=\frac{\rho UD}{\mu}=\frac{1.2\times15\times0.05}{1.8\times10^{-5}}=5.0\times10^{4}$$

This is in the subcritical range (about $10^3$ to $2\times10^5$), where the laminar boundary layer separates near $80^\circ$ and $C_D\approx1.2$, so the given value is reasonable.

(b) Drag force and power

Reference area (projected) $A=DL=0.05\times2=0.10$ m².

$$F_D=\tfrac12\rho U^2C_DA=0.5\times1.2\times15^2\times1.2\times0.10=16.2\ \text{N}$$

Power to overcome drag:

$$P=F_DU=16.2\times15=243\ \text{W}$$

(c) Vortex shedding

$$f_s=\frac{St\,U}{D}=\frac{0.2\times15}{0.05}=60\ \text{Hz}$$

Alternating vortices (von Karman street) produce an oscillating lift force at this frequency, which can excite vibration and noise.

(d) Bluff versus streamlined bodies

 Bluff body                 Streamlined body
   ____                        ______
  /    \  large wake        ---/      \___  thin wake
 |  O   |:::::::::::::       ---\______/
  \____/  low pressure         attached flow

• On a bluff body the boundary layer separates early because of the strong adverse pressure gradient. A broad low-pressure wake forms behind it, so pressure is much lower on the rear than the front: large pressure (form) drag dominates (friction drag is small).
• On a streamlined body the flow stays attached up to the tail, the wake is thin, and drag is mainly skin friction, which is much smaller.

CFD prediction: a 2D/3D domain extending about 10D upstream and 20D downstream is meshed with prism layers; RANS (k-$\omega$ SST) or unsteady URANS/LES is used; $C_D$ is computed by integrating surface pressure and shear stress over the body. Wake resolution and separation prediction are the main sources of error, so validation with experimental $C_D$ is needed.

Answer: $Re=5\times10^4$; $F_D=16.2$ N; $P=243$ W; $f_s=60$ Hz.

Evaluating forces from CFD

The aerodynamic force on a body is the integral of the surface pressure and wall shear stress over its surface $S$:

$$\vec F=\oint_S\left(-p\,\hat n+\vec\tau_w\right)dS$$

where $\hat n$ is the outward unit normal. The force is resolved relative to the free-stream direction:

• Drag $D$: component parallel to the free stream, made up of pressure (form) drag and skin friction drag.
• Lift $L$: component perpendicular to the free stream.

Non-dimensional coefficients:

$$C_L=\frac{L}{\tfrac12\rho U^2S},\qquad C_D=\frac{D}{\tfrac12\rho U^2S}$$

with $S$ = planform area (chord $\times$ span). In practice, the CFD code sums the force on each wall face and reports $C_L$ and $C_D$ through "force monitors"; the angle of attack is set by the flow direction. Accuracy requires a fine boundary-layer mesh ($y^+\approx1$ for k-$\omega$ SST), correct transition and separation modelling, and a mesh-independence check. A far-field distance of 10 to 20 chords is used.

Numerical

Planform area $S=0.5\times2=1.0$ m². Dynamic pressure:

$$q=\tfrac12\rho U^2=0.5\times1.2\times40^2=960\ \text{Pa}$$ $$L=C_LqS=0.9\times960\times1.0=864\ \text{N}$$ $$D=C_DqS=0.012\times960\times1.0=11.52\ \text{N}$$ $$\frac LD=\frac{C_L}{C_D}=\frac{0.9}{0.012}=75$$

Answer: $L=864$ N, $D=11.5$ N, $L/D=75$.