Flow Over Cylinder in Cross Flow
Evaluate convective heat transfer and aerodynamic drag for cylinders under external cross-flow conditions.
Boundary Layer Separation
As fluid passes over a circular cylinder, the boundary layer grows under a favorable pressure gradient on the front side, then faces an adverse pressure gradient on the back, causing flow separation. The separation angle shifts from $\approx 80^\circ$ (laminar separation) to $\approx 140^\circ$ (turbulent separation) at higher Reynolds numbers ($\text{Re}_D \gtrsim 2 \times 10^5$).
Parameters Setup
Results & Curves
📈 Nusselt number vs Velocity
Engine Terminal Output
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FLOW OVER CYLINDER IN CROSS FLOW
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--- INPUTS ---
Cylinder Diameter D = 0.050000 m
Free Stream Velocity V = 5.0000 m/s
Free Stream Temp T_inf = 25.00 C
Surface Temp T_s = 100.00 C
Film Temperature T_f = 62.50 C
--- FLUID PROPERTIES ---
Density rho = 1.1770 kg/m3
Viscosity mu = 1.8500E-05 Pa.s
Conductivity k = 0.026300 W/mK
Prandtl Pr = 0.7100
--- RESULTS ---
Reynolds Number Re_D = 1.5905E+04
Nu Churchill-Bernstein = 69.55
Nu Hilpert = 68.00
Convection Coeff h = 36.5829 W/m2K
Heat Transfer Q/L = 430.98 W/m
Drag Coefficient Cd = 0.4476
Drag Force F/L = 0.3292 N/m
BL Separation ~ 80 deg (laminar)
--- VELOCITY SWEEP ---
V[m/s] Re_D Nu_CB h[W/m2K] Q/L[W/m]
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0.100 3.181E+02 9.06 4.767 56.15
0.959 3.049E+03 28.38 14.928 175.86
1.817 5.781E+03 39.81 20.938 246.67
2.676 8.512E+03 49.09 25.820 304.18
3.534 1.124E+04 57.23 30.102 354.63
4.393 1.397E+04 64.64 33.999 400.54
5.252 1.671E+04 71.52 37.621 443.21
6.110 1.944E+04 78.01 41.035 483.44
6.969 2.217E+04 84.19 44.286 521.73
7.828 2.490E+04 90.12 47.402 558.44
8.686 2.763E+04 95.83 50.407 593.84
9.545 3.036E+04 101.36 53.316 628.12
10.403 3.309E+04 106.74 56.144 661.43
11.262 3.583E+04 111.98 58.899 693.89
12.121 3.856E+04 117.09 61.592 725.61
12.979 4.129E+04 122.11 64.227 756.66
13.838 4.402E+04 127.02 66.812 787.11
14.697 4.675E+04 131.85 69.351 817.02
15.555 4.948E+04 136.59 71.848 846.44
16.414 5.221E+04 141.27 74.307 875.41
17.272 5.494E+04 145.88 76.730 903.96
18.131 5.768E+04 150.42 79.121 932.13
18.990 6.041E+04 154.91 81.482 959.94
19.848 6.314E+04 159.34 83.815 987.42
20.707 6.587E+04 163.73 86.122 1014.60
21.566 6.860E+04 168.07 88.404 1041.48
22.424 7.133E+04 172.36 90.663 1068.10
23.283 7.406E+04 176.62 92.901 1094.46
24.141 7.680E+04 180.83 95.118 1120.59
25.000 7.953E+04 185.01 97.316 1146.48
--- CORRELATIONS ---
Churchill-Bernstein (Re*Pr>0.2):
Nu = 0.3 + 0.62*Re^0.5*Pr^(1/3)/[1+(0.4/Pr)^(2/3)]^0.25
* [1+(Re/282000)^(5/8)]^(4/5)
Ref: Incropera Ch.7 Eq.7.54
Calculation Methodology
Mathematical Model & Theory
Convection over a circular cylinder is determined by boundary layer development. The average Nusselt number is computed using two primary models:
1. Churchill-Bernstein Correlation (All $\text{Re}_D \text{Pr} > 0.2$):
$$\text{Nu}_D = 0.3 + \frac{0.62 \text{Re}_D^{1/2} \text{Pr}^{1/3}}{\left[1 + (0.4/\text{Pr})^{2/3}\right]^{1/4}} \left[1 + \left(\frac{\text{Re}_D}{282,000}\right)^{5/8}\right]^{4/5}$$
2. Hilpert Correlation:
$$\text{Nu}_D = C \text{Re}_D^m \text{Pr}^{1/3}$$
where $C$ and $m$ are constants evaluated based on the Reynolds number range.
Aerodynamic drag force per unit length is computed as:
$$F_D/L = \frac{1}{2} C_d \rho V^2 D$$
Academic References:
- Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer. 7th Edition, John Wiley & Sons.
- Churchill, S. W., & Bernstein, M. (1977). A Correlating Equation for Forced Convection from Gases and Liquids to a Circular Cylinder in Crossflow. J. Heat Transfer.
Worked Engineering Example
Water at $20^\circ\text{C}$ flows at $1.0\text{ m/s}$ perpendicular to a circular tube of diameter $25\text{ mm}$ with surface temperature maintained at $80^\circ\text{C}$. Calculate the convective heat transfer coefficient.
Step-by-step Solution:
1. Evaluate properties at film temperature $T_f = (20+80)/2 = 50^\circ\text{C}$:
- Water: $\rho = 997\text{ kg/m}^3, \mu = 8.9 \times 10^{-4}\text{ Pa·s}, k = 0.613\text{ W/m·K}, Pr = 6.13$.
2. Calculate Reynolds number:
$$\text{Re}_D = \frac{\rho V D}{\mu} = \frac{997 \times 1.0 \times 0.025}{8.9 \times 10^{-4}} = 28,005$$ 3. Evaluate Nusselt number via Churchill-Bernstein correlation:
$$\text{Nu}_D = 0.3 + \frac{0.62 \times (28,005)^{1/2} \times (6.13)^{1/3}}{\left[1 + (0.4/6.13)^{2/3}\right]^{1/4}} \left[1 + \left(\frac{28,005}{282,000}\right)^{5/8}\right]^{4/5}$$ $$\text{Nu}_D = 0.3 + \frac{0.62 \times 167.3 \times 1.83}{1.037} \times 1.17 = 227.1$$ 4. Calculate convection coefficient $h$:
$$h = \frac{\text{Nu}_D k}{D} = \frac{227.1 \times 0.613}{0.025} = 5,572\text{ W/m}^2\text{K}$$