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Validation of the computations of drag coefficient for the laminar flow over a flat plate at zero angle of inclination
December 17, 2021
1 Introduction
ANSYS Fluent is a simple computational tool for the study of fluid mechanics problems. The drag coefficient on a flat plate at zero degree angle of inclination in the presence of a laminar flow is computed using ANSYS Fluent and validated by comparison with the results from Blasius solution[Schlichting and Gersten(2003)].
2 Numerical methodology
The equations which govern the laminar flow over the flat plate are essentially derived from the continuity and momentum equations, given by equations 1 and 2, where ρ is the density of the fluid, v is the velocity vector, Sm is the source of mass, p is the pressure, τ is the stress tensor, g is the acceleration due to gravity and F is the external body force. Specifically for the steady, laminar flow over a flat plate with zero pressure gradient, the governing equations are equations 3 and 4, with the boundary conditions that the velocities, u = v = 0 on the wall and u = U∞, far-away from the wall [Schlichting and Gersten(2003)]. The mesh of the computational domain with the boundary conditions are given in figure 2. Find the mesh used in the computations attached here. The length of the flat plate is 0.5 m, air is the working fluid having a constant density, ρ = 1.226 kg/s. Inlet has the velocity component in the horizontal direction. Symmetry condition, where there is no velocity shear is given ahead of the flat plate. The flat plate has wall boundary condition and the outlet is given the pressure outlet boundary condition. Two dimensional, pressure based, coupled solver is used to solve the steady, laminar flow in ANSYS Fluent. Convergence of the residuals and the drag force with the iteration number is shown in figure 2.
3 Results and discussion
The results of computation of steady, laminar flow with the free-stream velocity, V∞ = 1 m/s are presented first. The velocity vectors within the boundary layer are shown in figure 3, with the length of the vectors and the color values representing the local magnitude of the velocity. The contours of velocity for the steady, laminar flow with the velocity of 1 m/s over the flat plate is shown in figure 3. Away from the thin boundary layer in the vicinity of the wall, the velocity is uniformly constant at the value of 1 m/s, shown in red color. The pressure is uniform everywhere in the two dimensional domain as observed in figure 3, as expected for the laminar flow over a flat plate with zero pressure gradient. The velocity profile in the wall normal direction, at several stream-wise distance namely, x = 0.1m, x = 0.2, . . , x = 0.5m are given in figure 3. The variation of skin friction coefficient from the leading edge of the flat plate for V∞ = 1m/s is given in figure 7. The velocity profiles within the boundary layer for the streamwise position x = 0.1 from the leading edge of the flat plate are shown in figure 8 in the dimensional form. The velocity profile in figure 8 is non-dimensionalized and compared against the numerically approximated profiles from the solution of Blasius equation in figures 9, where η = y(u∞/νx)^0.5 The frictional coefficient based on Blasius solution for a flat plate wetted on one side is cf= 0.664/(Rex)^0.5, where Rex is the local Reynolds number, which when compared against the findings from the solution in ANSYS Fluent from the leading edge of the flat plate is in good agreement as observed in figure 10. Drag values exerted on the flat plate for various free-stream velocities were computed from ANSYS Fluent and non-dimensionalized and plotted in figure 11 and compared with the drag coefficient based on Blasius solution namely, cf= 1.328/(Rel)^0.5, where l is the length of the plate [Schlichting and Gersten(2003)]. The comparison between the drag coefficients obtained based on CFD and Blasius solution is tabulated in table 1, a good agreement is achieved between the two, as observed in the form of small values of % error in the table.
4 Conclusion
Drag coefficient on a flat plate in a laminar flow is computed using ANSYS-Fluent and compared with the values obtained based on Blasius solution. Velocity contours, pressure contours, velocity distribution within the boundary layer, skin friction coefficient are obtained. A good agreement between the computed drag coefficients, non-dimensional velocity profiles and skin friction coefficient distribution along the length of the flat plate and the results from Blasius solution is obtained. Performance of the commercial software package ANSYS-Fluent in solving the simple case of steady, laminar flow field around a flat plate at zero incidence is exemplary.
References
[Schlichting and Gersten(2003)] Herrmann Schlichting and Klaus Gersten.
Boundary-layer theory. Springer Science & Business Media, 2003.
Re Drag [N] Drag CFD [N] cd cd CFD % error
3.38 E+04 0.00221 0.00226 0.00722 0.00739 2.3
6.77 E+04 0.00626 0.00635 0.00510 0.00518 1.45
1.02 E+05 0.01150 0.01162 0.00417 0.00421 1.06
1.35 E+05 0.01770 0.01785 0.00361 0.00364 0.83
1.69 E+05 0.02474 0.02490 0.00323 0.00325 0.67
2.03 E+05 0.03252 0.03270 0.00295 0.00296 0.55
2.37 E+05 0.04098 0.04117 0.00273 0.00274 0.46
2.71 E+05 0.05007 0.05026 0.00255 0.00256 0.39
Table 1: Comparison of drag: Blasius vs CFD
Equation 1: Continuity equation
Equation 2: Momentum equation
Eqn 3: Continuity
Eqn 4: Momentum equation
Figure 1: Mesh and the boundary conditions
Figure 2: Convergence of residuals and drag force
Figure 3: Velocity vectors, V∞ = 1 m/s
Fig 4: Velocity contours, V∞ = 1 m/s
Fig 5: Pressure contours, V∞ = 1 m/s
Fig 6: Velocity distribution in the boundary layer, V∞ = 1 m/s
Fig 7: Skin friction coefficient along the length of the flat plate at V∞ = 1 m/s
Fig 8: Velocity distribution at the station x = 0.1 m
Fig 9: Non-dimensional velocity distribution, V∞ = 1 m/s
Fig 10: Comparison of friction coefficient along the flat plate against Blasius solution
Fig 11: Comparison of drag coefficient against the results from Blasius solution at various Reynolds numbers
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