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 = 0.100000 m
Free Stream Velocity V = 0.3000 m/s
Free Stream Temp T_inf = 30.00 C
Surface Temp T_s = 150.00 C
Film Temperature T_f = 90.00 C
--- FLUID PROPERTIES ---
Density rho = 870.0000 kg/m3
Viscosity mu = 5.0000E-02 Pa.s
Conductivity k = 0.140000 W/mK
Prandtl Pr = 500.0000
--- RESULTS ---
Reynolds Number Re_D = 5.2200E+02
Nu Churchill-Bernstein = 114.24
Nu Hilpert = 100.12
Convection Coeff h = 159.9425 W/m2K
Heat Transfer Q/L = 6029.69 W/m
Drag Coefficient Cd = 1.1542
Drag Force F/L = 4.5189 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 1.740E+02 65.58 91.816 3461.39
0.148 2.580E+02 79.97 111.955 4220.59
0.197 3.420E+02 92.20 129.081 4866.22
0.245 4.260E+02 103.05 144.265 5438.66
0.293 5.100E+02 112.90 158.063 5958.85
0.341 5.940E+02 122.01 170.809 6439.34
0.390 6.780E+02 130.51 182.719 6888.36
0.438 7.620E+02 138.53 193.947 7311.61
0.486 8.460E+02 146.14 204.601 7713.29
0.534 9.300E+02 153.41 214.768 8096.55
0.583 1.014E+03 160.36 224.510 8463.84
0.631 1.098E+03 167.06 233.882 8817.14
0.679 1.182E+03 173.52 242.924 9158.01
0.728 1.266E+03 179.77 251.671 9487.78
0.776 1.350E+03 185.82 260.153 9807.53
0.824 1.434E+03 191.71 268.394 10118.20
0.872 1.518E+03 197.44 276.414 10420.58
0.921 1.602E+03 203.02 284.233 10715.35
0.969 1.686E+03 208.48 291.866 11003.10
1.017 1.770E+03 213.81 299.327 11284.36
1.066 1.854E+03 219.02 306.628 11559.59
1.114 1.938E+03 224.13 313.779 11829.20
1.162 2.022E+03 229.14 320.791 12093.55
1.210 2.106E+03 234.05 327.673 12352.97
1.259 2.190E+03 238.88 334.431 12607.75
1.307 2.274E+03 243.62 341.073 12858.16
1.355 2.358E+03 248.29 347.606 13104.44
1.403 2.442E+03 252.88 354.035 13346.81
1.452 2.526E+03 257.40 360.366 13585.48
1.500 2.610E+03 261.86 366.603 13820.62
--- 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 $h$:
$$h = \frac{\text{Nu}_D k}{D} = \frac{227.1 \times 0.613}{0.025} = 5,572\text{ W/m}^2\text{K}$$