Analysis and Simulation of Flat Plate Laminar Boundary Layer

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International Journal of Modern Research in Engineering & Management (IJMREM) ||Volume|| 1||Issue|| 9 ||Pages|| 51-55 || October 2018|| ISSN: 2581-4540

Analysis and Simulation of Flat Plate Laminar Boundary Layer 1, 1,2,3

Farzad Hossain, 2, Afshana Morshed, 3, Rifat Sultana

Department of Mechanical Engineering, Military Institute of Science and Technology, Dhaka, Bangladesh

----------------------------------------------------ABSTRACT-----------------------------------------------------In this paper, an analysis was done on laminar boundary layer over a flat plate. The analysis was performed by changing the Reynolds number. The Reynolds number was changed by changing horizontal distance of the flat plate. Since other quantities were fixed, the Reynolds number increased with increment of horizontal distance. Iterations were increased in scaled residuals whenever the Reynolds number was increased. Maximum value of velocity contour decreased with the increment of the Reynolds number. The value of the largest region of velocity contour decreased with the increment of the value of the Reynolds number and it also affected the appearance of contour. The value of pressure contour increased with the increment of the Reynolds number. Vertical distance versus velocity graph was not depended on the Reynolds number. In this graph, the velocity increased rapidly with the increment of vertical distance for a certain period. After that, the velocity decreased slightly with the increment of vertical distance. Finally, the velocity became around 1.05 m/s.

KEYWORDS: Laminar Boundary Layer, Flat Plate, Reynolds number, Velocity contour, Pressure contour ----------------------------------------------------------------------------------------------------------------------------- ---------Date of Submission: Date,11 October 2018 Date of Publication: Date 18. October 2018 ----------------------------------------------------------------------------------------------------------------------------- ----------

I.

INTRODUCTION

A viscous fluid may flow along a fixed inaccessible wall or exceed the rigid surface of a dipped body. An essential condition must be followed. It is obligatory to maintain zero velocity at any point on the wall or another fixed surface. The flow characteristic depends mainly upon the value of viscosity. The body may have streamlined shape and the viscosity may be small without being negligible. In this case, the modifying effect seems to be confined within narrow regions surrounding to the solid surfaces and these are known as boundary layers [1]. The fluid velocity changes quickly from zero to its main-stream value within such layers and a steep gradient of shearing stress can be implied [1]. As a result, it is impossible to neglect all the viscous terms in the equation of motion. The Reynolds number must be larger for achieving a well-defined laminar boundary layer. However, a breakdown of the laminar flow can be observed if the Reynolds number is extremely larger. At the local transition Reynolds number, laminar flow generally transformed into turbulent flow and for the flat plate, it should be in the range of 500,000 ≤ đ?‘…đ?‘’đ?‘?đ?‘&#x; ≤ 3,000,000 [2]. đ?œŒ Ă— đ?‘Ł Ă— đ?‘Ľđ?‘?đ?‘&#x; đ?‘…đ?‘’đ?‘?đ?‘&#x; = đ?œ‡ Here, đ?‘Ľđ?‘?đ?‘&#x; = the value of the horizontal distance, x where transition from laminar to turbulent flow occurs. When đ?‘Ľ < đ?‘Ľđ?‘?đ?‘&#x; , the flow is laminar and when đ?‘Ľ ≼ đ?‘Ľđ?‘?đ?‘&#x; , the flow is turbulent. Reynolds was the first person who made differentiation between laminar and turbulent flow and proposed a critical value of the Reynolds number, Re = 2100 for the upper limit of laminar flow in 1883 [3]. Moreover, Reynolds discovered new additional convection terms in turbulence in 1895 [4]. Since the terms have the units of stress, they are known as Reynolds stresses [4]. Kerswell considered both experimental and theoretical evidence points and found that the laminar flow state was linearly stable to any tiny disturbance [5]. Prandtl discovered boundary layer concept in 1904 [6]. Blasius discovered the solution of laminar flat plate boundary layer in 1908 [7]. Karman discovered momentum integral theory for flat-plate boundary layer in 1921 [8]. Previously, there was no computerized system to analyze the effects of flat flow laminar boundary layer with changing of the Reynolds number. This research gap has been fulfilled by this paper.

II.

METHODOLOGY

The analysis was made with the help of ANSYS Fluid Flow (Fluent). First of all, three rectangles were simulated. All of the rectangles had 0.5 m vertical dimension but horizontal dimensions were different. Four edges were considered, i.e. inlet, outlet, upper edge and lower edge. The energy equation was turned off and the flow was kept laminar. Air was used as a material whose density and viscosity were kept 1 kg/m 3 and 0.0001 kg/m-s respectively. Moreover, inlet velocity was kept 1 m/s.

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Analysis and Simulation of Flat Plate Laminar Boundary Layer

III. RESULTS AND DISCUSSION Reynolds number was calculated by using the formulađ?œŒĂ—đ?‘ŁĂ—đ?‘Ľ đ?‘…đ?‘’ = (1) đ?œ‡

Here, velocity (đ?‘Ł), density (đ?œŒ) and viscosity (đ?œ‡) were fixed. So, the Reynolds number (Re) depended on the value of x, which was the horizontal dimension of first, second and third rectangles. Since there was no vertical distance term in the equation, it had no effect on the Reynolds number. Table 1: Variation of Reynolds number (Re) with the variation of horizontal distance (x) Horizontal Distance (x) 0.8 m 1m 1.2 m

Reynolds Number (Re) 8000 10000 12000

It can be seen from the table that the Reynolds numbers was increased with the increasing of horizontal distance. Therefore, lower amount of horizontal distance gave lower amount of Reynolds number and higher amount of horizontal distance gave higher amount of Reynolds number.

Figure 1: Scaled Residuals (For Re = 8000)

Figure 2: Scaled Residuals (For Re = 10000)

Figure 3: Scaled Residuals (For Re = 12000)

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Analysis and Simulation of Flat Plate Laminar Boundary Layer

Variations can be seen in scaled residuals with Re = 8000, 10000 and 12000. For Re = 8000, around 272 iterations were done. For Re = 10000, around 291 iterations were done. For Re = 12000, around 307 iterations were done. So, iterations were increased whenever the Reynolds number was increased.

Figure 4: Velocity Contour (For Re = 8000)

Figure 5: Velocity Contour (For Re = 10000)

Figure 6: Velocity Contour (For Re = 12000) Some variations were seen by analysing three velocity contours. For Re = 8000, maximum value of velocity contour = 1.091e+000 m/s. For Re = 10000, maximum value of velocity contour = 1.087e+000 m/s. For Re = 12000, maximum value of velocity contour = 1.083e+000 m/s. So, maximum value of velocity contour decreased with the increment of the Reynolds number. Moreover, variations were seen in the shape of velocity contour. Comparatively large region contained maximum value of velocity contour when Re = 8000. However, comparatively small region contained maximum value of velocity contour when Re = 12000. Furthermore, value of the largest region of velocity contour was 1.034e+000 m/s when Re = 8000. The value of the largest region of velocity contour decreased with the increment of the value of the Reynolds number. Even it affected the colour of velocity contour. So, colour of the largest region of velocity contour converted from dark orange to light orange when the Reynolds number increased from 8000 to 12000.

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Analysis and Simulation of Flat Plate Laminar Boundary Layer

Figure 7: Pressure Contour (For Re = 8000)

Figure 8: Pressure Contour (For Re = 10000)

Figure 9: Pressure Contour (For Re = 12000) Some variations were seen by analysing three pressure contours. The value of pressure contour increased with the increment of the Reynolds number. Since the velocity distribution was almost same for three Reynolds number, only one distribution has been simulated for velocity vector.

Figure 10: Velocity Vector (For Re = 10000)

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Analysis and Simulation of Flat Plate Laminar Boundary Layer

Figure 10 represents velocity boundary layer and the distribution of velocity.

Figure 11: Vertical distance versus Velocity According to the vertical distance versus velocity graph, the velocity was zero when the vertical distance was zero. The velocity increased rapidly with the increment of vertical distance. When the vertical distance was 0.2 m, the velocity became around 1.15 m/s. Then the velocity decreased slightly with the increment of vertical distance. When the vertical distance was 0.5 m, the velocity became around 1.05 m/s.

IV. CONCLUSION Horizontal distance has an effect on the Reynolds number. Since the vertical distance has no effect on the Reynolds number, vertical distance versus velocity graphs are same for all of the Reynolds number. Moreover, the Reynolds number effects the scaled residuals, velocity contour and pressure contour.

REFERENCES [1] [2] [3]

[4] [5] [6] [7] [8]

K. P. Burr, T. R. Akylas and C. C. Mei, Two-Dimensional Laminar Boundary Layers, I-campus project School-wide Program on Fluid Mechanics Modules on High Reynolds Number Flows, 2016, 1-33. F. M. White, Fluid Mechanics, McGraw-Hill, 1999. O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistances in parallel channels, Phil. Trans. Roy. Soc. London, 1883, 174, 935-982. O. Reynolds, On the Dynamical theory of Incompressible Viscous Fluids and the Determination of the Criterion, Phil. Trans. Roy. Soc. London, 1895, 186A, 123-164. Kerswell RR, Recent progress in understanding the transition to turbulence in a pipe, Nonlinearity, 2005, 18(6), R17-R44. John D. Anderson, Ludwig Prandtl’s Boundary Layer, Physics Today, 2005, 58(12), 42. H. Blasius, Grenzschichten in Flüssikeiten mit kleiner Reibung, Z. Angew. Math. Phys., 1908, 56, 1-37. T. von Kármán, Über laminare und turbulente Reibung, Z. Angew. Math. Mech., 1921, 1(4), 233-252.

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