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 = 0.020000 m
Free Stream Velocity V = 20.0000 m/s
Free Stream Temp T_inf = 20.00 C
Surface Temp T_s = 60.00 C
Film Temperature T_f = 40.00 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 = 2.5449E+04
Nu Churchill-Bernstein = 91.28
Nu Hilpert = 90.92
Convection Coeff h = 120.0341 W/m2K
Heat Transfer Q/L = 301.68 W/m
Drag Coefficient Cd = 0.4376
Drag Force F/L = 2.0603 N/m
BL Separation ~ 80 deg (laminar)
--- VELOCITY SWEEP ---
V[m/s] Re_D Nu_CB h[W/m2K] Q/L[W/m]
-----------------------------------------------------------
0.100 1.272E+02 5.81 7.645 19.22
3.545 4.511E+03 34.87 45.857 115.25
6.990 8.894E+03 50.28 66.119 166.17
10.434 1.328E+04 62.80 82.582 207.55
13.879 1.766E+04 73.83 97.087 244.01
17.324 2.204E+04 83.92 110.350 277.34
20.769 2.643E+04 93.34 122.736 308.47
24.214 3.081E+04 102.25 134.462 337.94
27.659 3.519E+04 110.78 145.670 366.11
31.103 3.958E+04 118.98 156.456 393.22
34.548 4.396E+04 126.91 166.892 419.44
37.993 4.834E+04 134.62 177.030 444.93
41.438 5.273E+04 142.14 186.912 469.76
44.883 5.711E+04 149.48 196.571 494.04
48.328 6.149E+04 156.68 206.032 517.82
51.772 6.588E+04 163.74 215.318 541.15
55.217 7.026E+04 170.68 224.446 564.09
58.662 7.464E+04 177.51 233.432 586.68
62.107 7.903E+04 184.25 242.288 608.94
65.552 8.341E+04 190.89 251.026 630.90
68.997 8.779E+04 197.46 259.655 652.58
72.441 9.218E+04 203.94 268.184 674.02
75.886 9.656E+04 210.36 276.620 695.22
79.331 1.009E+05 216.71 284.970 716.21
82.776 1.053E+05 223.00 293.240 736.99
86.221 1.097E+05 229.23 301.434 757.59
89.666 1.141E+05 235.41 309.558 778.00
93.110 1.185E+05 241.53 317.616 798.26
96.555 1.229E+05 247.61 325.612 818.35
100.000 1.272E+05 253.65 333.548 838.30
--- 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 $h$:
$$h = \frac{\text{Nu}_D k}{D} = \frac{227.1 \times 0.613}{0.025} = 5,572\text{ W/m}^2\text{K}$$