Flow Over A Flat Plate

Samuel C. Fisher

Mathematical Nomenclature

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Re Reynolds Number

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CD Drag Coefficient

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u x-component of velocity

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v y-component of velocity

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L Plate Length

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ρ Fluid Density

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µ Kinematic Viscosity

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u∞ Free Stream Velocity

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Pgage Gage Pressure

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FD Drag Force

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V Velocity

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A Frontal Area

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G.R. Grid Resolution

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C.T. Convergence Tolerance

Introduction The purpose of this report was to analyze the effects of the Reynolds Number and various other parameters on drag coefficient of a flat plate subject to steady, incompressible flow. The Reynolds number values that were used in the analysis were chosen specifically based on the geometry of the flat plate to avoid simulating unsteady flow. Problem Definition The flow scenario that was analyzed is shown below in figure 1.

Figure 1

Air was chosen as the fluid in the flow scenario. The characteristic plate length,

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L , the dynamic viscosity of air,

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µ , the fluid density of air,

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ρ , and the x-component of the fluid velocity,

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u , were input to the Fluent model in accordance with the desired Reynolds number for each calculation. Reynolds number, a non-dimensional parameter, is given below in equation 1.

(1)

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Re =ρ uL

µ

The characteristic plate length, dynamic viscosity, and fluid density were approximated as the values shown in table 1 in order to simplify the process of varying the Reynolds number in the analysis.

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L

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m( )

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µ

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kgm ⋅ s

⎛ ⎝ ⎜ ⎞

⎠ ⎟

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ρ

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kgm3

⎛ ⎝ ⎜ ⎞

⎠ ⎟

1 0.0001 1 Table 1

Fluent was used to generate the geometry of the model. A mesh was then created with a grid resolution of 300 (50x60) cells. Boundary conditions were specified at the inlet and outlet of the section of flow encompassed by the mesh:

a)

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u@ inlet

=1m /s b)

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υ@ inlet

= 0 c)

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Pgage @ outlet= 0

A “no slip” wall boundary condition was also specified along the plate: d)

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u@ plate

= 0 The Fluent model solved the conservation of mass and conservation of momentum equations, shown in equation 2 and 3, to calculate the flow scenario from the user defined inputs and boundary conditions.

(2)

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dρdt

+∇ • ρu( ) = 0

(3)

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∂∂t

ρ υ ( ) +∇ • ρ

υ υ ( ) = −∇p +∇ • τ( ) + ρ

g + F

The model iterated until a predefined residual was satisfied, at which point it was possible to record the results of the flow calculations. The output values for drag force and drag coefficient were then printed. Drag force and drag coefficient are given by equations 4 and 5.

(4)

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FD =12⎛

⎝ ⎜ ⎞

⎠ ⎟ ρCDV

2A

(5)

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CD =2FDρ V 2A

Results/Analysis The geometry and mesh created with Fluent is shown in figure 2. This geometry involves the flat plate and a 0.5 m region above it to analyze the flow in that region. The simulation was run and

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u , or “X Velocity” was plotted vs. boundary layer height in figure 3. The values for drag coefficient were manually printed to the command window from the Fluent results, also shown in figure 3. The curve plotted in figure 3 fairly accurately resembles the general form of a boundary layer curve.

Figure 2

Figure 3

The simulation was then run three times holding Re and the mesh grid resolution constant at

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Re =10,000 and

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G.R. = 50 × 60( ) = 3000 cells. In these simulations the convergence tolerance was varied to determine its effect on the drag coefficient. The drag coefficient is plotted against the convergence tolerance in figure 4. The trend associated with the plotted data suggests that the drag coefficient decreases as the convergence tolerance decreases. The relationship between drag coefficient and convergence tolerance appears to be exponential.

Figure 4

The simulation was then run three times holding Re and the convergence tolerance constant at

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Re =10,000 and

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C.T. =1×10−5 . In these simulations the mesh grid resolution was varied to determine its effect on the drag coefficient. The drag coefficient is plotted against the mesh grid resolution in figure 5. The trend associated with the plotted data suggests that the drag coefficient decreases as the mesh grid resolution decreases. The relationship between drag coefficient and mesh grid resolution appears to be linear.

Figure 5

The simulation was again run three times holding Re, the convergence tolerance, and the mesh grid resolution constant at

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Re =10,000 ,

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C.T. =1×10−5 , and

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G.R. = 50 × 60( ) = 3000 cells. In these simulations the far-field boundary was to be varied to determine its effect on the drag coefficient. The bias factor, which controls the ratio of “closest to the plate mesh division length” to “farthest from the plate mesh division length”, was varied to simulate increasing the distance of the far-field boundary. The drag coefficient is plotted against the bias factor in figure 6. The trend associated with the plotted data suggests that the drag coefficient decreases as the

bias factor decreases, and consequently as the far-field boundary distance increases. The relationship between drag coefficient and far-field boundary distance appears to be linear. With the exception of this simulation the bias factor was held constant at 70.

Figure 6

The simulation was run three final times holding the convergence tolerance and the mesh grid resolution constant at

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C.T. =1×10−5 and

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G.R. = 50 × 60( ) = 3000 cells. In these simulations the Reynolds number was varied to determine its effect on the drag coefficient. The drag coefficient is plotted against the Reynolds number in figure 7. The trend associated with the plotted data suggests that the drag coefficient decreases as the Reynolds number decreases. The relationship between drag coefficient and the Reynolds number appears to be exponential. Expressing the drag coefficient as a function of Reynolds number using the equation of the curve in figure 7 yields:

(6)

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CD = f Re( ) = 0.0145 Re( )−1.714

Figure 7

Appendix A: Data

Drag Coefficient Vs. Convergence Tolerance Convergence Tolerance Drag Coefficient

1.00E-03 0.013714215  

1.00E-04 0.013679235  

1.00E-05 0.01367679  

1.00E-12 0.013676535

Drag Coefficient Vs. Grid Resolution Mesh Grid Resolution Drag Coefficient

3000 0.01367679  

1575 0.013425677  

750 0.013182538  

Drag Coefficient Vs. Bias Factor

Bias Factor Drag Coefficient 70 0.01367679  

15 0.013498397  

5 0.01318946  

Drag Coefficient Vs. Reynolds Number

Reynolds Number Drag Coefficient 10000 0.01367679  

5000 0.005231777 2500 0.001993318