1 Department of Energy Sciences, LTH MMV11 Fluid Mechanics LABORATION 1 Flow Around Bodies OBJECTIVES (1) To understand how body shape and surface finish influence the flow-related forces () To understand how scaling experiments can be used to determine the forces on a fullscale, real body (3) To understand the basic mechanisms of flow separation (4) To investigate pressure distributions around a circular cylinder and a wing profile SUMMARY The flow-related force vector acting on an immersed body can be divided into three named components, a drag (drag force), which acts in the flow direction, a lift (lift force) and a side force, all perpendicular to each other. The lift usually is in the direction so that it does a useful job, for instance upwards for an airplane in horizontal flight or downwards for inverted wings on race cars. In many cases the (time-mean) side force is zero, for instance when there is flow symmetry about the plane of lift and drag, as for an airplane flying in still air. Further, the components can be divided up with respect to their origin, wall surface pressure and wall friction. The pressure component of the drag, the pressure drag, is often referred to as the form drag since it is strongly dependent on the body form (shape). The remaining part is the friction drag, which is due to shearing viscous forces along the body surface. Flow similarity laws are crucial for model testing experiments. For instance, the Reynolds similarity law says that for incompressible flow about two geometrically similar bodies, without any effects of free surfaces, the flow itself is similar, if tested at the same Reynolds number. The lab session is based on three tests, in air flow. Test 1: Drag measurements Determine the drag for some different axisymmetric bodies and two circular cylinders. Test : Measurements of forces on an airfoil Determine lift and drag for an airfoil model at different angles of attack. Test 3: Pressure measurements Measure the distribution of static wall pressure around a circular cylinder in cross flow and an airfoil model, respectively. Also determine the form drag of the cylinder. PREREQUISITES Read this PM and pages , , , in F. M. White, Fluid Mechanics, 6th Edition 1, 006. REPORT Each student should account for all measurement data and results in the handed-out lab protocol, the report, which is to be finalized during the lab session. Time in lab: approx. 4 hours 1 Pages , , , in 5th edition; further page references are to the 6th edition. C. Norberg, 16/01/010
2 Forces on an immersed body A solid body that is exposed to flow of a viscous fluid is also affected by a flow-induced force. This force is the resultant of the normal and shear forces that acts locally on the body surface. The normal surface force is completely dominated by pressure action, the shear forces are due to viscous stresses only, and are normally referred to as friction forces. Consider a wing-like body of large width in relation to its chord, see Fig. 1. The flow around this body can be considered to be essentially two-dimensional. Figure 1: Forces on an immersed body, two-dimensional flow. The force vector F is divided into two components, the drag D, which acts in the flow direction, and the lift L, which acts normal to the flow. The drag itself is divided into pressure or form drag and friction drag. Form drag is due to the pressure forces acting on the surface (static pressure p); the friction drag is due the surface viscous shearing (frictional) forces (wall shear stress τ). The absolute value of F is normally expressed using a dimensionless coefficient C, defined as F ρ V = C A (1) V - oncoming, free stream velocity A - characteristic area ρ - density of fluid For geometrically similar bodies and for incompressible flows without influences of free surface effects we have the Reynolds similarity law: Flows around geometrically similar bodies are similar for equal Reynolds numbers The Reynolds number, Re, is dimensionless and is defined as Re ρv l V l = = () μ ν V - characteristic velocity, e.g. as above l - characteristic length μ - dynamic viscosity, ν - kinematic viscosity ( ν = μ / ρ ) For a certain geometrical flow situation, the Reynolds similarity law (or rule) means that the dimensionless coefficient C only is a function of the Reynolds number and a dimensionless
3 3 time (e.g. tv / l ), but only if the prerequisites are fulfilled. Thus, the following applies for the time-averaged force F in a stationary flow situation: C = f (Re) (3) If the a flow force on a body is to be determined from experiments, the Reynolds similarity law means that it is not necessary to use the same velocity, body size, fluid medium, fluid temperature, etc. as in the actual flow situation. If only the Reynolds number is the same, or if the Reynolds number is within an interval where C is a constant, the force from the scaled (model) situation can be transferred directly to determine the force on the actual body (fullscale, prototype). For instance, it is possible to transfer results from experiments in air to fullscale water conditions, but only if Reynolds number are equal. Drag The drag component caused by normal forces on the body, the form drag drag due to tangential forces D f, sum up to the total drag, D p, and the friction D = D p + D f (4) Consider Fig. 1 where ϕ is the angle between the surface normal n, at the surface element da, and the oncoming free stream velocity (of magnitude V). The pressure drag then is D p = p da cosϕ = ( p + ρ gz) da cosϕ = A A ρ V p *( x*, y*, z*,re) l da*cosϕ = A * ρ V l p *( x*, y*, z*,re) da*cosϕ = ρ V l f1(re) A * Dimensionless quantities are denoted with a *, e.g. x = x / l. Now introduce the projected area of the body in the free stream direction, or any other characteristic area of the body, and denote it by A. Since this area is proportional to l, it follows that * D p D p ρ V = CD, p A, where C D, p, the pressure drag coefficient, only depends on Re. The total friction drag becomes D f = τ da sinϕ = μ V t V Vt * da sinϕ = μ l n da sinϕ l n = μ μ V l f(re) = ρ V l f(re) = ρ V l f3(re) ρ V l *
4 4 Thus, Since D f ρ V = CD, f A, where the friction drag coefficient C D, f only depends on Re. = C C we then obtain for the total drag: C D D, f + D, p D = C D A ρ V where the (total) drag coefficient C only depends on Re, (Re). Lift The total lift force is determined in a similar fashion, D C D L = C L A ρ V where the lift coefficient C L only depends on Re, C L (Re) Force components measured in a model scale experiment can thus be transferred to any other scale, if there is full geometrical similarity and equal Reynolds numbers (and of course, stationary, incompressible flow without any influence of free surface, e.g., wave interactions). The choice of characteristic area and length For geometrical similar bodies the characteristic length l and the characteristic area A can be chosen arbitrarily. However, as a general rule for bluff bodies, bodies that under normal conditions give rise to large separated wake regions (for instance a sphere, ordinary cars, etc.), the characteristic area is normally chosen as the projected area of the body when viewing it from the upstream, along the free stream velocity; the so-called frontal area. As a characteristic length it is customary to choose a typical cross-stream dimension within the frontal area. For cylinders of circular cross-section the characteristic length is chosen as the diameter d, the frontal, characteristic area then is A = b d, where b is the cylinder length. Also for spheres the characteristic length is chosen as the diameter ( A = π d / 4 ). For slender bodies of the wing type ( wing profiles ) the characteristic length is chosen as the mean chord, ( l = c ), see Fig.. The so-called planform area (projected wing area) is used as characteristic area, A = Ap = bc, where b is the wing span. Figure : Wing profile at angle of attack α.
5 5 Symmetrical bodies In the mean sense, a fully symmetric body, e.g. a sphere, only has a force component in the flow direction, the drag D. According to eq. (1) this drag corresponds to a drag coefficient C D, which under the conditions of Reynolds similarity law only is a function of the Reynolds number (Re). The variations of C D with Re, for the flow around a sphere and a long circular cylinder in cross-flow, are shown in Fig. 3 (smooth surfaces). At very low Reynolds numbers for the sphere the drag coefficient in Fig. 3a approaches a straight line. For low enough Reynolds numbers, see p. 480 in White, the drag can be determined from the Stokes formula, D = 3π μ Vd (5) According to eq. (1) the drag coefficient becomes C D = 4 Re which means a straight line in logarithmic diagram (dotted line in Fig. 3a). 5 Within 5 10 < Re < 3 10 the drag coefficient for a sphere is approximately constant ( = 0.44 ± 0.07 ). Thus the drag within this interval is approximately proportional to the C D velocity squared, D V. At low Re (approx. Re < 1), from Stokes formula, it is proportional to the velocity, D V. Figure 3a: Drag coefficient for a smooth sphere (Fox & McDonald 1994). At high Re the surface roughness has a significant influence on C D, see Fig. 5.3 and D5. in White.
6 6 Figur 3b: Drag coefficient for a long and smooth circular cylinder in cross-flow. We recall that the total drag can be divided into form (pressure) drag and friction (viscous shear) drag. The Reynolds number is a measure of the ratio between inertia forces (mass times acceleration) and viscous forces in the flow field. This means that the viscous forces at large Re become very small in relation to the pressure forces, with pressure forces approximately balancing the inertial forces. Consequently, at high enough Re the form drag dominates, and it can be determined solely from the pressure field acting on the body surface. Fig. 3b shows that the form drag dominates the total drag for the circular cylinder as from about Re = For the cylinder, the friction drag coefficient can be approximated as, C D, f 3.5/ Re (6) C D, f For the circular cylinder and within 3 10 < Re < 3 10, C D = 1.08 ± At Re = the friction part is about 16% of the total drag, at Re = 3 10 it is only about 0.5%. 5 At Re 3 10, for both the sphere and the cylinder and with increasing Re, there is a dramatic decrease in the drag coefficient. In fact, over a short interval, the total drag actually diminishes with increasing velocity. The phenomenon is called drag crisis, and it depends on a change of character for the boundary layer, which influences the separation process; see 6 pages in White. At higher Re, for Re > 4 10 (approx.), the drag coefficient for the cylinder again settles to an approximate constant level, C = 0.6 ± 0.1, see Fig. 3b. 5 D
7 7 4 Figure 4: Pressure distribution round a circular cylinder in cross-flow ( Re 10 ). Fig. 4 shows a schematic of the pressure distribution along the stagnation streamline, for the circular cylinder in cross-flow at relatively high Reynolds numbers (where the drag coefficient is approximately constant, C D 1). The pressure difference Δp is referred to the undisturbed pressure far upstream, in this case the atmospheric pressure, Δp = p patm. Due to deceleration, there is a pressure rise in front of the cylinder. At high Re, the viscous forces in this region are negligible; the pressure rise at the stagnation point then is equal to the dynamic pressure ρv /. Due to acceleration the pressure decreases on the frontal side. Boundary-layer separation occurs at ϕ 80 ; the pressure in the separated (wake) region is approximately constant. Downstream of the cylinder the pressure eventually recovers to the undisturbed value. The pressure distribution around the cylinder surface will be studied further during the laboratory exercise. Wing profiles (aerofoils) Under normal circumstances a wing is subjected not only to a drag but also a lift. Consider the flow around a wide-span wing (a wing profile or aerofoil) exposed to a constant velocity (constant Reynolds number, assumed high), at varying angles of attack α, see Fig.. Since α is part of the geometry both CD and C L will depend on α. Starting at small α, the drag coefficient is approximately constant but then starts to increase; the lift coefficient increases rapidly, almost linear with α. The lift coefficient reaches a maximum at a critical angle of attack, α = αc ; beyond this angle there is a rapid decrease in C L while there is a continuing increase of C D. The occurrence of lift can be explained as follows. At small angles of attack the flow is attached to wing almost to the trailing edge. Due to this viscous adhesive capacity there is a net deflection of the flow downwards 3, which according to the laws of Newton means that there is an opposing force upwards on the wing. The wing profile acts like a turning vane. On the frontal part of the upper side of the wing there is a compression of streamlines, the velocity increases and this means, at high enough Re, that the pressure is lowered (cf. Bernoulli equation). The upper side is thus often called the suction side. On the lower side the streamlines diverge somewhat and the pressure increases slightly (pressure side). At high Re the occurrence of lift is thus due mainly to the pressure difference between the upper and lower side. On the upper side, however, the velocity eventually decreases and the pressure 3 Also the upstream flow is affected, at the wing tip flow is deflected upwards and this also contributes to lift.
8 8 thus starts to increase. This is an adverse pressure gradient and it means that there is a potential for flow separation, and for high enough angles of attack this certainly will happen, starting close to the trailing edge. This is about the point of maximum lift ( α = αc ). A slight increase in α then will move the position of flow separation rapidly upstream, the flow deflection diminishes significantly and there is a rapid drop in lift. Eventually the separation is so massive that the flow is barely deflected at all, the force is then dominated by (pressure) drag; see Fig. 5 and Fig. 7.4 in White. Beyond the critical angle α c the wing is said to be stalled; α c is often called the stall angle. The phenomenon of stall is of great importance when landing an airplane, as it is then crucial to have high enough lift with much drag, at a velocity as low as possible. If stall occurs the loss in lift might result in a too steep approach of the landing-ground. Figure 5: Polar diagram for a wing with b / c = 5 (Finnemore & Franzini 00). The angle of attack for which the ratio between lift and drag is a maximum is called the (best) gliding angle ( α g ). The angle of attack at cruising conditions is normally set at around this value. For a glider in still air, maximum sailing distance will occur at this angle of attack. For the cambered wing in Fig. 5, α 1. g End effects, wing-tip vortices All real wings (and cylinders) have limited width (length). Naturally, there will be flow distortions near the ends, end effects. If the width is high enough the end effects will be of minor importance, sometimes even neglected. At limited widths, the end effects can be reduced by using end plates, thin plates mounted at the ends, along the flow and of special design. For real finite-width wings there will be an overflow from the pressure (lower) side of
9 9 the wing to the suction (upper) side. At the wing tips the pressure is equalized (zero lift). The overflow results in so-called wing-tip vortices, which reduces the lift. It also increases the drag. Since this drag is due to lift it is usually called lift-induced drag. For more details, see Chapter 8.7 in White. Experimental equipment Wind tunnel During scientific measurements it is required to have a flow that is highly uniform, unidirectional and non-turbulent. To achieve this in air it is usually recommended to use a closed-return wind tunnel, with a plenum chamber with several gauze screens followed by a well-designed contraction before the test section, driven by low-noise, highly effective fan unit. However, such wind tunnels are very expensive and require much space. During the lab exercise we will instead use two small (and a bit noisy!), open wind tunnels (fan units), which are quite sufficient for our purposes. The wind tunnels used in this lab basically consist of an axial fan within a circular duct followed by a short contraction (nozzle). One unit has a guiding vane ring that will dampen the flow distortions due to the rotating fan blades. This unit also has exchangeable exit nozzles (of different outlet diameter). Prandtl tube (Pitot-static tube) The free stream velocity V is measured by using a so-called Prandtl tube, also called Pitotstatic tube, see pages 404/5 in White. Along a streamline in stationary, incompressible and frictionless flow the Bernoulli equation applies. Along a horizontal streamline or a streamline where effects of gravitation are negligible, it reads V p + ρ = p0 = const. (7) where p is static pressure, ρ the fluid density and V is the local velocity. The combination ρv / is called the dynamic pressure, the difference between the pressure in a zero-velocity state (stagnation pressure p 0 ) and the flowing state along the same streamline. At the tip of the Prandtl tube there is a pressure hole; it also has pressure holes along the perimeter further downstream. When pointing towards the flow, the frontal tip will become a stagnation point ( V = 0, p = p t ). At the position for the perimeter holes, and by careful design, the pressure has recovered exactly to the undisturbed value upstream, the free stream pressure p. If friction effects can be neglected, as they indeed can for high enough Reynolds numbers, the pressure difference between these two positions is equal to the dynamic pressure, since p = by eq. (7). The undisturbed velocity along the stagnation streamline then is t p 0 ( p 0 p) V = (8) ρ The air density can be calculated from the ideal gas law, p ρ =, where R = 87 J/(kg K) (9) RT
10 10 Pressure and force measurements Measurements of pressure differences will be carried out using a differential liquid U-tube manometer, with readings directly in pascal (Pa). A more detailed description and how to use it will be given by the instructor. Drag and lift components are measured using a twocomponent force balance, to be described by the instructor. Outline The laboratory session consists of three parts: 1. Measurements of Drag (axisymmetric bodies) (a) Measure the drag D on a set of axisymmetric bodies (Fig. 6), at a certain air velocity. Figure 6: Axisymmetric bodies. (b) Measure the drag D on two circular cylinders in cross-flow, for two or three velocities. Analysis (1) Compute Re and C D for all cases () Compare the results with Fig. 3 in this PM, and Table 7.3 in White. Measurements of drag and lift (wing profile) For a certain velocity, measure drag D and lift L for a wing profile at different angles of attack, e.g., α = -5 o, - o, 0 o, o, 5 o, 10 o, 15 o, 0 o, 5 o. Analysis (1) Compute the Reynolds number, C D, C L and the lift-to-drag ratio L / D = C L / C D. Why is L / D called the glide ratio? (think about unpowered descent of a plane in still air) What is the gliding angle α g? () Plot C D and C L as a function α (same diagram). What is the approximate stall angle? What happens at stall?
11 11 3. Pressure measurements (circular cylinder and wing profile) (a) Pressure distribution around a circular cylinder The surface pressure is measured through a small drilled hole in the surface. Measure the pressure difference Δp ϕ between the surface pressure (at mid-span) and the ambient pressure at various angles from the scale, e.g., -0 o, -10 o, 0 o, 10 o, 0 o, 40 o,.. 80 o, 90 o, 100 o, 10 o,..., 180 o, -0 o. The angular position for the actual stagnation point (where ϕ = 0 ) may not be identical to the scale value. Actual angles can be determined from the (assumed) symmetry condition. The pressure difference Δp ϕ is measured using a differential micro-manometer. Analysis The component that contributes to the pressure drag is stagnation point, The form drag D p Δ p ϕ cosϕ. Since ϕ = 0 is the pϕ = 0 = ρv / (Bernoulli equation, high Re). per unit width is determined from the integration around the perimeter, Δ π D p = 0 Δ p ϕ cosϕ R dϕ Because of symmetry it is enough to integrate over half the perimeter and multiply with, The form drag coefficient is π Δp cosϕ = R Δpϕ cosϕ dϕ = ρv R dϕ (10) Δp D p 0 0 ϕ= 0 π C D, p = D ρ V p R = π Δpϕ cosϕ dϕ Δp 0 ϕ= 0 (11) (b) Pressure measurements along a wing profile Pressure differences Δp x between the wall static pressure at different fixed positions and the ambient pressure are measured; see Fig. 7. The pressure tap at x = 0 is assumed to be at the stagnation point (limited angles of attack). Figure 7: Pressure holes on the wing profile.
12 1 Analysis Compute Reynolds numbers for both the cylinder and the wing profile. (a) Circular cylinder (1) Plot C p ( ϕ) = Δpϕ / Δpϕ = 0 and f ( ϕ) = ( Δpϕ / Δpϕ = 0 )cosϕ = C p cosϕ in the same diagram. Which part of the cylinder contributes most to the drag, frontal side or rear side? () Determine the pressure drag coefficient, C D, p, graphically. (3) Estimate the total drag coefficient, C D = CD, p + CD, f, and compare with results from previous total drag measurements and Fig. 3b in this PM. (b) Wing profile (1) Plot f ( x) = Δpx / Δpx= 0 = Δpx / Δp1, where x is the abscissa for the projected holes; see Fig. 7. Which side contributes most to the lift, the upper side or the lower side? On which surface is there a risk for flow separation? Why? () Estimate the lift coefficient C L REFERENCES Finnemore, E. J. & Franzini, J. B. (00), Fluid Mechanics (with Engineering Applications), Tenth Edition, McGraw-Hill. Fox, R. W. & McDonald, A. T. (1994), Introduction to Fluid Mechanics, Fourth Edition, John Wiley & Sons, Inc. White, F. M. (006), Fluid Mechanics, Sixth Edition, McGraw-Hill.