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 = 0.025000 m
Free Stream Velocity V = 1.0000 m/s
Free Stream Temp T_inf = 20.00 C
Surface Temp T_s = 80.00 C
Film Temperature T_f = 50.00 C
--- FLUID PROPERTIES ---
Density rho = 997.0000 kg/m3
Viscosity mu = 8.9000E-04 Pa.s
Conductivity k = 0.613000 W/mK
Prandtl Pr = 6.1300
--- RESULTS ---
Reynolds Number Re_D = 2.8006E+04
Nu Churchill-Bernstein = 216.99
Nu Hilpert = 197.90
Convection Coeff h = 5320.5952 W/m2K
Heat Transfer Q/L = 25072.71 W/m
Drag Coefficient Cd = 0.4359
Drag Force F/L = 5.4318 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 2.801E+03 60.71 1488.648 7015.09
0.269 7.533E+03 102.96 2524.509 11896.47
0.438 1.226E+04 134.79 3305.140 15575.11
0.607 1.700E+04 162.16 3976.089 18736.88
0.776 2.173E+04 186.88 4582.226 21593.23
0.945 2.646E+04 209.81 5144.422 24242.52
1.114 3.119E+04 231.42 5674.383 26739.90
1.283 3.592E+04 252.02 6179.436 29119.91
1.452 4.066E+04 271.80 6664.526 31405.84
1.621 4.539E+04 290.91 7133.170 33614.27
1.790 5.012E+04 309.46 7587.976 35757.50
1.959 5.485E+04 327.53 8030.937 37844.90
2.128 5.958E+04 345.17 8463.617 39883.86
2.297 6.432E+04 362.45 8887.269 41880.27
2.466 6.905E+04 379.40 9302.917 43838.96
2.634 7.378E+04 396.06 9711.405 45763.92
2.803 7.851E+04 412.46 10113.444 47658.48
2.972 8.324E+04 428.61 10509.636 49525.49
3.141 8.798E+04 444.56 10900.497 51367.38
3.310 9.271E+04 460.30 11286.473 53186.25
3.479 9.744E+04 475.85 11667.952 54983.93
3.648 1.022E+05 491.24 12045.274 56762.02
3.817 1.069E+05 506.47 12418.740 58521.94
3.986 1.116E+05 521.56 12788.617 60264.94
4.155 1.164E+05 536.51 13155.143 61992.15
4.324 1.211E+05 551.33 13518.532 63704.58
4.493 1.258E+05 566.03 13878.976 65403.13
4.662 1.306E+05 580.61 14236.648 67088.62
4.831 1.353E+05 595.09 14591.707 68761.80
5.000 1.400E+05 609.47 14944.296 70423.34
--- 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 $h$:
$$h = \frac{\text{Nu}_D k}{D} = \frac{227.1 \times 0.613}{0.025} = 5,572\text{ W/m}^2\text{K}$$