EXTERNAL INCOMPRESSIBLE VISCOUS FLOW
External flows are flows over bodies immersed in an unbounded fluid. The flow over semi-infinite plate, and the flow over a sphere are examples of external flows. Some external flows that are of interest to engineers are shown in the figure.
Figure 1. Examples for external flow
Characteristics of Flow Past an Object
Character of the steady, viscous flow past a flat plate and a cylinder (Munson)
- Flow is dominated by inertia effects (viscous effects are negligible) everywhere except the region near solid surface (boundary layer).
- As the Reynolds number is increased (by increasing U, for example), the region in which viscous effects are important becomes smaller in all directions except downstream.
- The flow separates from the body at the separation location BOUNDARY LAYER CONCEPT The concept of boundary layer was first introduced by Ludwig Prandtl, a German aerodynamicist, in 1904.
Prior to Prandtl’s historic breakthrough, the science of fluid mechanics had been developing in two different directions: Theoretical hydrodynamics evolved from the solution of Euler’s equations of motion for inviscid flows in 1755. Since the results of hydrodynamics contradicted many experimental observations, practicing engineers developed their own empirical art of hydraulics.
Navier-Stokes equations describing the motion of a viscous flows were developed by Navier in 1827 and independently by Stokes in 1845. Theoretically, all flow problems can be analyzed by solving Navier-Stokes equations. However, the mathematical difficulties in solving Navier-Stokes equations, except for simple flow cases, prohibit the theoretical treatment of viscous flows.
Prantdtl showed that many viscous flows can be analyzed by dividing the flow field into two regions: One close to the solid boundaries, the other covering the rest of the flow field.
Only in thin region adjacent to a solid boundary, (the boundary layer) the viscous effects are important. In this region, the viscous forces are much more significant than the inertia forces. The velocity of the fluid at the wall relative to the solid boundary is zero and increases toward the main stream. Shear stresses in this zone are very high owing to the existence of extremely high velocity gradients at and near the solid boundaries.
In the region outside the boundary layer the influence of viscosity is negligible, so that the inertial forces dominate viscous ones. Therefore, the flow in this region may be considered inviscid, and the potential flow theory may be used for analyzing the flow.
ME304 7 2 Figure 2. Boundary Layer on a Flat Plate
Near the leading edge of the plate, there is a laminar boundary layer flow and grows in thickness. A transition region is reached where the flow changes from laminar to turbulent. The transition depends on the Reynolds number which is defined as Ux Ux Reor Re For calculation purposes, under typical flow conditions, transition usually is considered to occur at Re= 500,000 Therefore if;
Re ≤ 500,000 flow is laminar
Re > 500,000 flow is turbulent
This boundary layer characteristic occurs in a variety of flow situations, not just on flat plates. For example, boundary layers form on the surfaces of cars, in the water running down the gutter of the street, and in the atmosphere as the wind blows across the surface of the earth (land or water).
ME304 7 3 a) Boundary Layer Thickness
The boundary layer thickness, δ, is usually defined as the distance from the solid boundary to the point where the velocity is within 1 percent (or u=0.99U) of the freestream velocity.
u= 0.99U
u=0.99U at y=δ
Figure 3. Definition of boundary layer thickness. b) Boundary Layer Displacement Thickness The boundary layer displacement thickness, δ*, may be defined as the distance by which the solid boundary would have to be displaced to maintain the same mass flow rate in an imaginary frictionless flow.
The decrease in mass flow rate due to the influence of viscous forces is; U u wdy 0
U w U u wdy 0 For incompressible flow, =constant, solving for *, we get displacement thicness as uu 11 dy dy Since u≅ U at y= δ, the integrand is essentially zero for y≥ δ. 00UU ME304 7 4 c) Boundary Layer Momentum Thickness, q Flow retardation within boundary layer also results in a reduction in momentum flux at a section compared with inviscid flow.
The momentum deficiency through the boundary layer is u U u wdy 0 In the case of ideal flow, to obtain this momentun deficiency, it will be necessary to move the solid boundary upward by q. The momentum deficiency, due to the displacement of solid boundary by q, would be U2qw. Thus we can write, U2 q w u U u wdy 0 For incompressible flow, =constant, solving for q, we get momentum thickness as
u u u u q 11 dy dy Since u≅ U at y= δ, the integrand is essentially zero for y≥ δ. 00UUUU
ME304 7 5 Example: (Fox) A laboratory wind tunnel has a test section that is 305 mm square. Boundary-layer velocity profiles are measured at two cross-sections and displacement thicknesses are evaluated from the measured profiles. At section ①, where the * freestream speed is U1= 26 m/s, the displacement thickness is δ1 = 1.5 mm. At * section ②, located downstream from section ①, δ2 = 2.1 mm. Calculate the change in static pressure between sections ① and ②. Express the result as a fraction of the freestream dynamic pressure at section ①. Assume standard atmosphere conditions.
GIVEN: Flow of standard air in laboratory wind tunnel. Test section is L= 305 mm * * square. Displacement thicknesses are δ1 = 1.5 mm and δ2 = 2.1 mm. Freestream speed at section ① is U1= 26 m/s.
FIND: Change in static pressure between sections ① and ②. (Express as a fraction of freestream dynamic pressure at section ①.)
Assumptions: Solution: (1) Steady flow. (2) Incompressible flow. (3) Flow uniform at each section outside δ*. (4) Flow along a streamline between sections ① and ②. (5) No frictional effects in freestream. (6) Negligible elevation changes.
The idea here is that at each location the boundary-layer displacement thickness effectively reduces the area of uniform flow, as indicated in the following figures: * * Location ② has a smaller effective flow area than location ① (because δ2 > δ1 ). Hence, from mass conservation the uniform velocity at location ② will be higher. Finally, from the Bernoulli equation the pressure at location ② will be lower than that at location ①.
To be completed in class
ME304 7 6 ME304 7 7 BOUNDARY LAYER FLOW ANALYSIS
In theory, incompressible, viscous flows over any object can be analyzed by solving the governing Navier-Stokes equations. However, there is no analytical solution to these equations for flows over any shaped object. Hence, by using the boundary layer concept, the governing equations are simplified and solved for the flow in the boundary layer.
There are two general approaches for the analysis of the flow in the boundary layers:
1) Exact solution (Simplified Navier-Stokes equations are solved) 2) Approximate solution (Momentum integral equation is solved)
LAMINAR FLAT-PLATE BOUNDARY LAYER: EXACT-SOLUTION (Blasius Boundary Layer Solution)
Consider steady, two dimensional, and laminar flow of an incompressible fluid over a flat plate with zero pressure gradients under the absence of body forces. In theory, the details of viscous incompressible flow past any object can be obtained by solving the governing Navier–Stokes equations. For steady, two-dimensional laminar flows with negligible gravitational effects, Navier Stokes equations for two-dimensional flow field are u u1 p 22 u u uv 22 x y x x y
22 u= 0.99U v u1 p v v uv 22 x y y x y The conservation of mass equation for incompressible flow is uv 0 xy By using boundary layer concepts, Prandtl was able to impose certain approximations (valid for large Reynolds number flows) and thereby to simplify the governing equations. p 22 u u Note that 0 and in the boundary layer x x22 y Hence equations become Boundary conditions: u u 2 u At wall y0 u 0, v 0 uv x y y2 du At y u U,0 dy ME304 7 8 In 1908, H. Blasius, one of Prandtl’s students, was able to solve these simplified equations for the boundary layer flow past a flat plate parallel to the flow. From the Blasius solution, the boundary layer thickness, boundary layer displacement thickness, momentum thickness and wall shear stress are obtained.
To solve boundary layer equations analytically, a nondimensional variable and a nondimensional stream function are defined, and the boundary layer equations are reduced to an ordinary differential equation.
ME304 7 9 at y=0, η=0 f= df/dη= 0 (u=0, v=0) at y=, df/dη= 1 (u=U)
ME304 7 10 ME304 7 11 Example: Air with a density of 1.2 kg/m3 and kinematic viscosity of 1.5x10-5 m2/s is flowing over a flat plate at a uniform velocity of 0.3 m/s. The length and width of the flat plate are 4.5 mm and 0.5 m, respectively. a) Determine the variation of the boundary layer thickness, the boundary layer displacement thickness, the boundary layer momentum thickness over the flat plate. Also plot their variation. b) Determine the variation of the shear stress over the flat plate. c) Determine the wall shear stress coefficient, Cf.
SOLUTION
To be completed in class
ME304 7 12 ME304 7 13 ME304 7 14 APPROXIMATE ANALYSIS OF BOUNDARY LAYER FLOW
MOMENTUM INTEGRAL EQUATION In the previous section, an exact solution of the boundary layer equation is obtained for steady, laminar flow of an incompressible fluid over a flat plate with zero pressure gradient. However, it is not always possible to obtain the exact solutions of the boundary layer equations for more complicated flow fields. For this reason some approximate solution methods are derived for obtaining the boundary layer thickness. One of these approximate solution methods is the Von Karman integral momentum equation, which gives very good results to find boundary layer thickness, not only in the laminar flow range but also in the turbulent flow range.
To derive the momentum integral equation, consider incompressible, steady, two dimensional flow over a solid surface. For our analysis we choose a differential control volume, of length dx, width w and height , as shown in the figure.
Assumptions:
1) Steady flow
2) Two-dimensional flow
Continuity Equation:
For steady flow,
For the above control volume, in terms of mass flow rate through each face, it can be expressed as.
ME304 7 15 Let us evaluate these terms for the differential control volume.
Surface ab is located at x. Since flow is two-dimensional, the mass flux through ab is
ME304 7 16 ME304 7 17 ME304 7 18 ME304 7 19 at y=0, u=0 at y= δ, u=U
at y= δ,
at y=0,
3) Determine the unknown coefficients of assumed velocity profile using the above boundary conditions.
4) Substituting velocity profile into momentum integral equation and carrying out the integrals, obtain an ordinary differential equation for boundary layer thickness, .
5) Solve the differential equation for boundary layer thickness, .
ME304 7 20 ME304 7 21 ME304 7 22 ME304 7 23 ME304 7 24 ME304 7 25 ME304 7 26 ME304 7 27 Example: Water with a density of 1000 kg/m3 and kinematic viscosity of 1x10-6 m2/s is flowing over a flat plate. Water approaches to the flat plate at a uniform velocity of 0.5 m/s. The flat plate is 8 m long and 2 m wide. Determine the friction drag on one side of the flat plate.
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