Masters in Mechanical Engineering. Problems of incompressible viscous flow. 2µ dx y(y h)+ U h y 0 < y < h,


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1 Masters in Mechanical Engineering Problems of incompressible viscous flow 1. Consider the laminar Couette flow between two infinite flat plates (lower plate (y = 0) with no velocity and top plate (y = h) with velocity equal to U) at a distance h. a) Establish the boundary conditions of the problem. b) Determine the velocity component normal to the plates, v. c) Check that a velocity profile given by: u = 1 dp 2µ dx y(y h)+ U h y 0 < y < h, with dp dx is the longitudinal component of the pressure gradient, µ is the dynamic viscosity of the fluid and y is the normal distance to the plates is an exact solution of the boundarylayer equations. 2. The velocity profile of a laminar boundarylayer over a flat plate is approximated by a parabola. u U e = Aη + Bη 2 with η = y δ, where u is the velocity component parallel to the wall, U e is the external velocity, y is the distance to the wall, δ is the boundarylayer thickness and A and B are constants. a) Determine the constants A and B. b) Determine the (dimensionless) displacement thickness, δ /δ, momentum thickness, θ/δ, and the shape factor H. c) Using the integral equation of Von Kármán, determine the (dimensionless) change of the boundarylayer thickness along the plate, δ/x. (x is the distance along the wall to the beginning of the boundarylayer). d) Determine the evolutions of δ /x, θ/x and of the skin friction coefficient, C f = τ w / 1 2 ρu2 e, along the plate (τ w is the shearstress at the wall and ρ is the fluid density). 1
2 3. The velocity profile of a laminar flat plate boundarylayer is given by u ( π ) ( π ) = Asen U e 2 η + Bcos 2 η com η = y δ, where u is the velocity component parallel to the wall, U e is the external velocity, y is the distance to the wall, δ is the boundarylayer thickness and A and B are constants. a) Determine the constants A and B. b) Determine the (dimensionless) displacement thickness, δ /δ, momentum thickness, θ/δ, and the shape factor H. c) Using the integral equation of Von Kármán, determine the (dimensionless) change of the boundarylayer thickness along the plate, δ/x. (x is the distance along the wall to the beginning of the boundarylayer). d) Determine the evolutions of δ /x, θ/x and of the skin friction coefficient, C f = τ w / 1 2 ρu2 e, along the plate (τ w is the shearstress at the wall and ρ is the fluid density). 4. A velocity profile of a laminar flat plate boundarylayer is approximated by a 3 rd order polynomial that includes four constants, A, B, C and D. u U e = Aη 3 + Bη 2 +Cη + D com η = y δ, where u is the velocity component parallel to the wall, U e is the external velocity, y is the distance to the wall and δ is the boundarylayer thickness. Determine the values of A, B, C and D. 5. The velocity profile (u is the velocity component parallel to the wall, U e is the external velocity, y is the distance to the wall and δ is the boundarylayer thickness) u U e = 1 e 4.605η with η = y δ is a smooth curve that satisfies u = 0 at y = 0 and u = 0.99U e at y = δ, which might suggest that it could be a reasonable description of the velocity profile of a laminar flat plate boundarylayer. However, assuming this profile and using the integral equation of Von Kármán one obtains δ x 9.2(Re x) with Re x = U ex ν,
3 which is nearly 100% different from the analytical solution of Blasius (with u = 0.995U e for y = δ) δ x 5(Re x) 0.5. What is the reason for this discrepancy? 6. Air at 20m/s and 20 o C (ν = m 2 /s) flows over a flat plate. Assume that the critical Reynolds number (Re crit ) is a) Estimate the distance to the leading edge of the plate, x, for which the boundarylayer thickness, δ, is equal to 1mm. b) Estimate the distance to the leading edge of the plate, x, for which the boundarylayer thickness, δ, is equal to 10mm. c) For a plate with a length, L, of 50cm, determine the resistance force per unit width assuming that the flow is laminar for Re x < Re crit and turbulent for Re x > Re crit. Determine the same force for fullyturbulent flow. d) Repeat item c) for a plate with L =2m. 7. Assuming a dimensionless form of the velocity profile, it is possible to obtain approximate solutions of the boundarylayer equations using the integral equation of Von Kármán. For the velocity profile: u U e = A+Bsen ( π 2 η ) +Csen 2( π 2 η ) with η = y δ where u is the velocity component parallel to the wall, U e is the external velocity, y is the distance to the wall, δ is the boundarylayer thickness and A, B and C are constants, a) Determine the most appropriate values of A, B and C. b) Determine the parameter λ = θ 2 du e ν dx for the flow separation profile (τ w = 0) of this family. c) For a zero pressure gradient boundarylayer, for which regime (laminar or turbulent) is the proposed family of profiles appropriate? Justify your choice. 3
4 1 0 sen 2( π 2 η ) dη = 1 2, 1 0 sen 3( π 2 η ) dη = 4 3π 1 0 sen 4( π 2 η )dη = The velocity profile of an atmospheric turbulent boundarylayer (with δ 600m) is similar to the velocity profile of a flat plate boundarylayer (zero pressure gradient). Make the following assumptions: the surface is hydraulically smooth; the law of the wall is valid and at an height of 80m the velocity is 10m/s. In these conditions, determine: a) The shearstress at the surface. b) The velocity 1.7m and 0.17m above the ground. c) The thickness of the linear (laminar) sublayer. 9. Consider the flow around a NACA airfoil (figure 1). As a first approximation, assume that the boundarylayers develop in zero pressure gradient for an angle of attack equal to zero degrees. Assume that the transition from laminar to turbulent is instantaneous, i.e. it is concentrated in one point. Consider that the distance to the stagnation point can be measured along the chord, i.e. s x. a) Assume that the pressure drag coefficient (C Dp ) is independent of the Reynolds number (Re = U c ν ) and that the transition point for Re = and Re = is in the same location. Show that the transition point is located at ( ) x c b) Estimate C Dp for zero degrees of angle of attack. c) Estimate the dimensionless momentum thickness (θ/c) at the trailing edge at zero degrees of angle of attack for Re = d) Assume now that transition is forced at the leading edge. Estimate the airfoil drag coefficient for Re = Discuss the result obtained. 10. Consider a small aerodynamical tunnel with a flat bottom and an adjustable ceiling as illustrated in figure 2. Consider that the tunnel s width is sufficient to assume twodimensional flow. The velocity at the inlet is 4
5 h y Figure 1: NACA airfoil. α U e (x) L x Figure 2: Aerodynamic tunnel. uniform and equal to 25 m/s. The tunnel s length is L=1m and the height at the inlet is h=30cm. a) Neglecting the effect of the boundarylayers on the tunnel s walls, determine the distribution of the mean velocity as a function of the angle 5
6 α. b) Using Thwaites method, determine the angle α for which laminar flow separation occurs on the bottom wall at the tunnel s outlet x = L. c) For the conditions of question b), estimate the momentum thickness (θ) at the outlet (x = L) for the bottom wall. d) For the conditions of question b) and assuming the velocity profile given by U U e = C 1 +C 2 η +C 3 η 2 +C 4 η 3 com η = y δ, estimate the boundarylayer thickness (δ) at the outlet (x = L) for the bottom wall. e) Is the situation of question b) (laminar flow separation at the outlet) physically realistic? 11. Consider a bidimensional, incompressible, laminar boundarylayer developing in the x direction under a pressure gradient given by: dp dx = ρa2 x, where ρ is the fluid density and a is a constant. a) Determine the evolution of the momentum thickness, θ, with x. b) Determine the evolution of the shape factor, H, and displacement thickness, δ, with x. Discuss the result. c) Determine the evolution of the skin friction coefficient, C f, with x. 12. Table 1 presents measurements performed in a turbulent boundarylayer on a smooth wall. The fluid is air (assumed to be incompressible) with the following properties: ρ air = 1.25 kg/m 3 and ν air = m 2 /s. x stands for the distance to the leading edge of the location where the measurements were performed and y is the distance to the wall. Table 1 x=1.5m y (mm) U (m/s) Determine the wall shearstress (τ w ) and the thickness of the viscous sublayer. Explain in detail the required calculations. 6
7 13. Consider a bidimensional, incompressible, laminar boundarylayer developing in the x direction in a region where the external velocity is given by: U e = 20x 2 (m/s), a) Determine the evolution of the momentum thickness, θ, with x. b) Determine the evolution of the shape factor, H, and displacement thickness, δ, with x. Discuss the result. c) Determine the evolution of the skin friction coefficient, C f, with x. d) Assuming that the velocity profile is a cubic polynomial, estimate the boundarylayer thickness (δ) at x = 1m. Figure 3: NACA airfoil. 14. Consider the flow around a NACA airfoil (figure 3). As a first approximation, assume zero pressure gradient for the development of the boundarylayers at zero degrees angle of attack. Assume that transition is 7
8 instantaneous (critical Reynolds number equal to the transition Reynolds number) and that the distance to the stagnation point can be measured along the chord, i.e. s x. Neglect the pressure drag (form drag) of the airfoil. a) For a Reynolds number of , estimate the location of the transition point. b) Estimate C D when transition to turbulent flow is forced at the leading edge of the airfoil. c) For the conditions of question b) (fullyturbulent flow), estimate the momentum thickness on the wake of the airfoil for a sufficiently large distance to the trailing edge that guarantees p p. 15. A velocity profile for a twodimensional, incompressible boundarylayer is given by: U U e = Aln(η + 1)+Bη +C com η = y δ. where A, B and C are constants. a) Determine the constants A, B and C. b) To which flow regime is this velocity profile suited? Justify your answer. 1 0 ln(η + 1)dη = 2ln(2) 1, ηln(η + 1)dη = 0.25 (ln(η + 1)) 2 dη = 2(1 ln(2)) Consider a bidimensional, incompressible, laminar boundarylayer developing in the x direction in a region where the external velocity (U e ) is given by: U o U e (x) = (1+x/L) 2 and the momentum thickness, θ, is zero at x = 0. 8
9 (ν = m 2 /s) a) Determine the pressure gradient imposed to the boundarylayer. b) Using Thwaites method, determine the distance x/l for which the boundarylayer separates. c) Estimate the relation between U o and L that makes the previous estimate realistic. 17. A wind tunnel has a flat bottom wall with 1m length. The height of the top wall is given by ( y(x) = h 1+A x ) 2 h where y is the distance between the tunnel walls, x is the distance to the inlet along the bottom wall and A is a constant (figure 4). h = 0,2m is the height of the inlet section of the tunnel. The velocity profile at the inlet is uniform and the velocity magnitude is U o. Assume that the flow in the tunnel is steady, bidimensional and incompressible. = m 2 s 1, ρ air = 1.2 kg m 3 ) h U o y(x)=h(1+a(x/h)) 0 X 1 Figure 4: Wind tunnel. a) Neglecting the effect of the wall boundarylayers, estimate the pressure gradient along the tunnel. b) Determine the relation between the constant A and the distance for which the bottom wall boundarylayer separates, x sep. c) Estimate the relation between A and U o to obtain a realistic answer to the previous question. d) Estimate the value of y(x) to obtain a zero pressure gradient imposed to the wall boundarylayers, without neglecting its effect. 9
10 18. A publicity panel with an area of 4m 2 is located in a region where the wind speed is parallel to the panel. Admit that the wind speed is U o = 10m/s and that the length and width of the panel cannot be smaller than 1m. ν air = m 2 s 1, ρ air = 1.2 kg m 3 ) a) Determine the pressure gradient imposed to the boundarylayers on the panel surface. Justify your answer. b) Estimate the length and width of the panel that minimize the drag force of the panel.. c) Estimate the increase of the drag force if the wind speed becomes 25m/s. d) Explain the consequences for the previous calculations of having wind perpendicular to the panel. 19. A wind tunnel of L = 1m length as an adjustable top wall, h = f(x). At the inlet, the tunnel height is h o = 0.3m and the flow is uniform with a velocity magnitude of U o = 10m/s. Assume that the tunnel has smooth wall and that the flow is incompressible and bidimensional flow. = m 2 /s, ρ air = 1.2 kg/m 3 ) a) Neglecting the effect of the wall boundarylayers, determine the shape of the top wall of the tunnel, h = f(x), to obtain a pressure gradient imposed to the bottom wall boundarylayer given by dp dx = ρu2 o L ( 1+ x ) L b) Without neglecting the effect of the wall boundarylayers, determine the shape of the top wall of the tunnel, h = f(x), to obtain zero pressure gradient imposed to the bottom wall boundarylayer. c) For the conditions of question b), determine the drag force of the bottom wall of the tunnel. d) Estimate the maximum value of U o for which the solution technique of the previous question is valid. 20. A panel for publicity purposes is supposed to have an area of 4m 2. The panel will be placed in a location with the wind (wind speed U o ) aligned with the length of the panel. The smallest admissible length and width of. 10
11 the panel is 1m. Assume a Reynolds number at transition equal to = m 2 /s, ρ air = 1.2 kg/m 3 ) Assuming that the panel has zero thickness, determine the panel dimensions that minimize the drag force for the following conditions: a) Wind speed U o = 5m/s b) Wind speed U o = 10m/s c) Assuming that the panel thickness is not zero, but with negligible pressure drag, determine the panel dimensions that minimize the drag force for a wind speed of U o = 10m/s. d) For the conditions of question c), determine the longitudinal velocity component at the trailing edge of the panel for y = 0.1δ. 21. A foil with a 1m chord is supposed to operate in air or water for a uniform incoming flow with a velocity magnitude of U = 10m/s. At small angles of attack, one may assume that the boundarylayers develop in zero pressure gradient starting from a stagnation point located at the leading edge of the foil. Neglect pressure drag, i.e. foil drag equal to friction drag, and assume instantaneous transition from laminar to turbulent flow for a Reynolds number of = m 2 /s, ρ air = 1.2 kg/m 3 ) (ν water = m 2 /s, ρ water = 1000 kg/m 3 ) a) Which fluid (air or water) leads to the largest dissipated power? Give a clear justification to your answer. b)which fluid (air or water) leads to the largest momentum thickness at the trailing edge of the foil? Give a clear justification to your answer. c) Which fluid (air or water) leads to the largest change in the dissipated power when roughness is applied at the leading edge of the foil to force transition to turbulent flow? Give a clear justification to your answer. d) Estimate the velocity of the incoming flow U in water that leads to the same dissipated power of the foil operating in air with U = 10m/s. 22. A laminar airfoil exhibits a drag coefficient, C d, of at small angles of attack for a Reynolds number of Assume that the critical Reynolds number is equal to the transition Reynolds number (Re crit = Re trans = ) and that boundarylayers develop in zero pressure gra 11
12 dient. = m 2 /s, ρ air = 1.2 kg/m 3 ) a) Estimate the pressure (form) drag coefficient of the airfoil. b) Estimate the increase (in percentage) of the friction drag coefficient when transition to turbulent flow is forced at the leading edge of the airfoil. c) For the conditions of question b) (forced transition), the pressure (form) drag coefficient is smaller, equal or larger that the pressure (form) drag coefficient of question a) (natural transition)? Give a clear justification to your answer. d) Determine the limiting value of the momentum thickness in the wake for the conditions of question a). 23. A publicity panel is supposed to be placed on the top of a vehicle that moves at 20 km/h in a region without wind. The panel must have an area of 2m 2 and its length can not exceed 2.5m. Assume that the critical Reynolds number is equal to the transition Reynolds number (Re crit = Re trans = 10 6 ). = m 2 /s, ρ air = 1.2 kg/m 3 ) a) Assuming that the panel has zero thickness, determine the dimensions of the panel that minimize the dissipated power. Estimate the minimum dissipated power. b) If the panel has a finite (but small) thickness (for example 2mm), determine dimensions of the panel that minimize the dissipated power. Determine (in percentage) the change of dissipated power compared to the conditions of question a) (zero thickness panel). c) For the conditions of question b), the wind exhibits a small lateral(perpendicular to the panel) component of 2 km/h. Estimate the friction drag coefficient for such conditions. Give a clear justification to your answer. d) Estimate the dissipated power in the conditions of question c). Make all the assumptions that you deem to be necessary. 24. An airfoil with a 1m chord is supposed to operate a uniform incoming flow with a velocity magnitude of U = 15m/s. As a first approximation, assume zero pressure gradient for the development of the boundarylayers at zero degrees angle of attack and that the distance to the stagnation point can be measured along the chord, i.e. s x. Assume that transition occurs 12
13 instantaneously at 60% of the chord (x/c = 0.6). = m 2 /s, ρ air = 1.2 kg/m 3 ) (ν water = m 2 /s, ρ water = 1000 kg/m 3 ) a) Estimate the friction resistance coefficient of the airfoil. b) Estimate the increase (in percentage) of the drag coefficient when transition to turbulent flow is forced at the leading edge of the airfoil using roughness. Assume that the pressure (form) drag coefficient is 20% of the resistance coefficient of airfoil with and without roughness. c) If the same foil is operating in water, the change of the resistance coefficient with and without roughness is larger, equal or smaller than that obtained for air? Give a detailed justification to your answer. d) The drag coefficient of the airfoil (air flow) was predicted numerically with a grid suitable for the application of the no slip condition using wall functions. Is the same grid appropriate for the prediction of the drag coefficient of the foil in a water flow with the same wall boundary conditions? Give a detailed justification to your answer. 25. The NACA airfoil has a drag coefficient, C d, approximately equal to for 0.1 <C l < 0.5 at a Reynolds number, Re, of (figure 5). As a first approximation, assume zero pressure gradient for the development of the boundarylayers at zero degrees angle of attack. Assume that transition is instantaneous (critical Reynolds number equal to the transition Reynolds number) and that the distance to the stagnation point can be measured along the chord, i.e. s x. Consider that the pressure (form) drag is 13% of the drag coefficient of the airfoil. a) Estimate the location of the transition point. b) Assuming that C d is also approximately equal to for 0.1 < C l < 0.5 and Re = and considering that the transition Reynolds number is the same, estimate the pressure (form) drag of the airfoil for Re = c) Is the result of the previous question physically correct? Give a quantitative justification to your answer. (If you did not solve the previous questions assume that the pressure (form) drag is 30% of the airfoil drag at Re = and that the transition Reynolds number is ). 13
14 Figure 5: NACA airfoil. 0.3 Re L =U L/ν= Re L =U L/ν=10 9 U/U e U/U e E07 2E07 3E07 4E07 5E07 y/l y/l Figure 6: Mean velocity profile of a turbulent boundarylayer. 26. Figure 6 and table 2 present the mean velocity profile, U, at a given crosssection, x, of the flow over a flat plate at a Reynolds number of Re L = The profile is calculated numerically with RANS equations supplemented by the SST version of the twoequation eddyviscosity k ω model. The 14
15 Tabela 2 y/l U/U e calculations were performed without the use of wall functions, i.e. the shearstress at the wall was obtained directly from its definition. U e = U is the undisturbed velocity, L is the plate length, ν is the kinematic viscosity of the fluid and y is the distance to the wall. a) Determine the skin friction coefficient C f. b) Determine y + u τ y/ν at the point 10 8 L away from the wall, y = 10 8 L. c) Determine the Von Kármán constant, κ, from the velocity profile obtained numerically. d) Estimate the location, x/l, of the selected crosssection. 27. The aerodynamic characteristics of a NACA airfoil are given in figure 7. For angles of attack in the laminar bucket, it is possible to assume that the boundarylayers develop in zero pressure gradient and that the distance to the stagnation point can be measured along the chord. a) Assuming that the transition point is located at 55% of the chord for a Reynolds number (Re = U c/ν) of , estimate the pressure drag coefficient in the laminar bucket region. b) Estimate the increase of the airfoil drag coefficient for the same range of angles of attack when transition to turbulent flow is forced at the leading edge. Discuss the result using the data available in figure 2. c) For the conditions of question b) (turbulent flow starting at the leading edge), estimate the maximum nondimensional thickness (use the airfoil chord as the reference length) of the region where the Reynolds stresses are negligible. d) What is the importance of the value obtained in the previous question for the numerical calculation of the flow around the airfoil? 28. Figure 8 presents the aerodynamic characteristics of a NACA air 15
16 Figure 7: NACA airfoil. foil obtained experimentally. For the laminar bucket region, 0 < C l < 0.4, for which the drag coefficient remains approximately constant, it is possible to assume that the boundarylayers develop in zero pressure gradient and that the distance to the stagnation point can be measured along the chord. For a Reynolds number of and for the laminar bucket region: a) Estimate the friction drag coefficient when transition is forced at the leading edge. b) Estimate the pressure drag coefficient when transition is forced at the leading edge. c) Assuming that the pressure drag coefficient does not change when transition is forced at the leading edge, estimate the location of the transition point for natural transition in the laminar bucket region. d) Using the same assumptions of the previous questions, is it possible to 16
17 make the same estimates of drag coefficients for a Reynolds number of ? Give a clear justification of your answer. Figure 8: Perfil NACA Consider a finite wing with a plan area of 12m 2 and mean chord, c, of 2m that is going to operate in a small aircraft at the cruise speed of 180km/h. A 1:4 model was tested in a wind tunnel that produces a flow at 20m/s. = 1, m 2 /s, ρ air = 1,2 kg/m 3 ) a) Should transition to turbulent flow be forced at the leading edge of the model? Give a clear justification of your answer. At zero lift angle, assume that the boundarylayers develop in zero pressure gradient and that the pressure drag coefficient corresponds to 5% of the friction drag coefficient. b) Estimate the drag coefficient of the model for the zero lift angle (according to the answer of question a)). 17
18 c) Estimate the drag coefficient of the wing. d) A RANS solver including an eddyviscosity turbulence model is available for the determination of the drag coefficients of the model and of the wing. The calculations should apply directly the noslip condition at the wall, i.e. no wall functions. Is it possible to use the same grid (equal values of x/c and y/c) for the flows around the model and the wing? Give a clear justification of your answer. 30. An airfoil with a 1m chord is going to operate in an uniform flow of air. At small angles of attack, it is possible to assume that the boundarylayers develop in zero pressure gradient and that the stagnation point is at the leading edge. Assume that transition from laminar to turbulent flow occurs instantaneously for a Reynolds number of = 1, m 2 /s, ρ air = 1,2 kg/m 3 ) a) Estimate the velocity of the incoming flow knowing that transition is located at 60% of the chord (x/c = 0.6). b) Estimate the friction drag coefficient of the airfoil. c) Estimate the friction drag coefficient of the airfoil when roughness is applied at the leading edge of the airfoil. d) The pressure drag coefficient (C d ) pressure with transition at 60% of the chord is larger, smaller or equal than (C d ) pressure with roughness applied on the leading edge of the airfoil? Give a clear justification to your answer. 31. An airfoil with a chord of 1,1m is going to operate in a uniform flow of air with a velocity of 150km/h. At small angles of attack, it is possible to assume that the boundarylayers develop in zero pressure gradient and that the stagnation point is at the leading edge. Assume that transition from laminar to turbulent flow occurs instantaneously for a Reynolds number of and that the pressure drag coefficient is equal to 10% of friction drag coefficient. = 1, m 2 /s, ρ air = 1,2 kg/m 3 ) a) Estimate the drag coefficient of the airfoil. b) Estimate the friction drag coefficient of the airfoil when roughness is applied at the leading edge of the airfoil. c) Suppose that the flow in the conditions of question a) is calculated using 18
19 the timeaveraged RANS equations supplemented with an eddyviscosity turbulence model. Estimate the distance of the first interior grid node to the wall (y 2 /c) to apply directly the noslip condition (without wall function). d) Is the distance determined in the previous question also suitable for the conditions of question b)? Give a clear justification to your answer. 32. Consider a finite wing of 12m 2 de plane area and mean chord c of 1m that is going to operate in a small aircraft with a cruise speed of 144 km/h. The wing section is a symmetric airfoil of small thickness. For zero lift angle, assume that the boundarylayers develop in zero pressure gradient and that the stagnation point is at the leading edge. Assume that transition from laminar to turbulent flow occurs instantaneously for a Reynolds number of 10 6 = 1, m 2 /s, ρ air = 1,2 kg/m 3 ) a) Estimate the drag friction coefficient of the airfoil b) Estimate the increase of the drag friction coefficient of the airfoil (in percentage) for a small angle of attack that already exhibits a suction peak close to the leading adge. c) The pressure drag coefficient in the conditions of question b) should be larger, smaller or equal than in the conditions of question a)? Give a clear justification to your answer. d) For the flow at zero angle of attack, estimate the distance of the first interior grid node to the wall (y 2 /c) to apply wall functions boundary conditions at the airfoil surface in the numerical solution of the timeaveraged RANS equations supplemented with an eddyviscosity model. 33. A rectangular publicity panel with an area of 3m 2 is located in a region where the wind (of speed U o ) is aligned with the length of the panel. The minimum length and width allowed is 1m. Assume that the transition from laminar to turbulent flow is instantaneous (Critical Reynolds=Transition Reynolds= ), and that the panel as no thickness. The goal of the exercise is to determine the dimensions of the panel that minimize the drag force at a given wind speed U o. = m 2 /s, ρ air = 1.2 kg/m 3 ) a) Determine the wind speeds U o for which the length should be larger than the width. 19
20 b) Determine the wind speeds Uo for which the length should be equal to 3m. c) Assuming that the thickness of the panel is not zero but the pressure drag is negligible, what changes in the solution of questions a) and b)? Give a clear justification of your answer. d) In the conditions of question c) and for a wind speed of Uo = 15m/s, determine the drag force of the panel. Figure 9: NACA airfoil. 34. Figure 10 presents the aerodynamic coefficients of the NACA airfoil at three different Reynolds numbers: 3 106, and At small angles of attack (laminar bucket) For zero lift angle, it is possible to assume that the boundarylayers develop in zero pressure gradient and that the stagnation point is at the leading edge. Assume that transition from laminar to turbulent flow occurs instantaneously and that the distance to the stagnation point (leading edge) may be measured along 20
21 the chord (neglecting the thickness). Consider that the ratio between the friction and pressure drag coefficients is constant. = 1, m 2 /s, ρ air = 1,2 kg/m 3 ) a) Assuming that for small angles of attack the transition point is located at 50% of the chord (x = 0.5c) for a Reynolds number of , estimate the ratio between the pressure and friction drag coefficients. b) Estimate the airfoil drag coefficient at small angles of attack and Reynolds number of with forced transition at the leading edge. Discuss the result based on the data presented in figure 9. c) The flow in the conditions of question b) is going to be calculated with the timeaveraged RANS equations supplemented with an eddyviscosity model. Estimate the minimum distance of the first interior grid node (y 2 /c) to the airfoil surface to apply wall functions boundary conditions at the airfoil surface. d) Is the distance determined in the previous question also suitable for the conditions of question a)? Give a clear justification to your answer. 35. Figure 10 presents the aerodynamic coefficients of the NACA 0012 airfoil at three different Reynolds numbers: , and For zero lift angle, it is possible to assume that the boundarylayers develop in zero pressure gradient and that the stagnation point is at the leading edge. Assume that transition from laminar to turbulent flow occurs instantaneously and that the distance to the stagnation point (leading edge) may be measured along the chord (neglecting the thickness). Consider that the ratio between the friction and pressure drag coefficients is constant. = 1, m 2 /s, ρ air = 1,2 kg/m 3 ) For zero angle of attack and Reynolds numbers of and the drag coefficient is equal to In these conditions, estimate: a) The location of transition point for each of the Reynolds numbers. b) The ratio between the pressure and friction drag coefficients. c) The flow at Reynolds number of is going to be calculated with the timeaveraged RANS equations supplemented with an eddyviscosity model. Estimate the maximum distance of the first interior grid node (y 2 /c) to the airfoil surface to apply directly the noslip condition at the 21
22 Figure 10: NACA 0012 airfoil. wall (without wall functions). d) Is the grid of the previous question suitable for the calculation of the flow at a Reynolds number of 108 with wall functions boundaryconditions at the airfoil surface? Give a clear justification of your answer. 36. a The purpose of this exercise is to determine the integral parameters of a turbulent boundarylayer over a flat plate using the Von K arm an integral equation. To this end, consider a mean velocity profile given by y 1/7 U = Ue δ where U is the mean velocity component parallel to the plate, Ue is the velocity of the outer flow, y is the normal coordinate to the wall and δ is the thickness of the boundarylayer. The LudwiegTillman correlation is used to determine the skin friction coefficient θ U e C f = H ν 22
23 where H is the shape factor of the mean velocity profile, θ is the momentum thickness and ν is the kinematic viscosity of the fluid. Assume that the flow is fully turbulent (no laminar region) and deliver all the results in dimensionless form choosing the most appropriate reference values. a) Determine the displacement thickness δ as a function of the distance to the leading edge of the plate x. b) Determine the drag coefficient C D for a plate of length L. b Consider now the numerical calculation of the same flow using the Reynolds timeaveraged boundarylayer equations with an eddyviscosity model. c) Determine the eddyviscosity ν t = µ t /ρ in the law of the wall region to guarantee that the mean velocity profiles satisfy the log law. U = 1 ( u τ κ ln uτ y ) +C ν where u τ is the friction velocity and κ and C are constants. d) If we choose the twoequation k ε model, what is the expected behaviour of k and ε in the log law region? 37. Consider a rectangular outdoor panel with an area of 3m 2. Assume that the panel has no thickness and that its length L has to be larger than 1m and can not exceed 3m, 1m L 3m. Admit that transition from laminar to turbulent flow occurs instantaneously (Re crit = Re trans = ). = m 2 /s, ρ air = 1.2 kg/m 3 ) Estimate the dimensions of the panel that lead to the maximum and minimum drag for the following conditions: a) 20 km/h wind parallel to the panel. b) 44 km/h wind parallel to the panel. c) 44 km/h wind with a small angle of attack (for example 2 o ). d) Describe how would you apply the boundary conditions on the surface panel if you want to calculate the drag force in conditions b) solving the Reynolds Averaged NavierStokes equations numerically. Estimate the distance to the panel surface y 2 /L of the first interior grid node. 23
Laminar Flow. Chapter ZERO PRESSURE GRADIENT
Chapter 2 Laminar Flow 2.1 ZERO PRESSRE GRADIENT Problem 2.1.1 Consider a uniform flow of velocity over a flat plate of length L of a fluid of kinematic viscosity ν. Assume that the fluid is incompressible
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