Flat Plate Equation Research Articles

Conflicting Aspects in the Flat Plate Boundary Layer Conventional Solution

The solution of the steady-state isothermal gas flow with zero-pressure gradient over a flat plate is classical and was presented by Blasius in 1908. Despite being reduced to a third-order Ordinary Differential Equation, nobody has solved it using just three boundary conditions. In consequence, the “solutions” (exact or numerical) are not unique, do not satisfy Prandtl’s boundary layer concept, and give rise to imprecise criteria of the boundary layer thickness definition (0.99.U∞, where U∞ is the free stream velocity) and to idealizations such as the displacement thickness, δ∗, and the momentum thickness, δI. Even though η, the similarity parameter, is defined as a function of x, it is surprising that it is seen as a constant, η∞, at the limit of the boundary layer, being valid for the entire plate. It is shown that uncountable “solutions” satisfying the classical equation and its three natural boundary layer conditions can be built. The reduction technique of the Boundary Value Problem, BVP, to an Initial Value Problem, IVP, and the Perturbation Analysis Technique, used as an attempt to incorporate a fourth boundary condition to the differential equation, are discussed. A more general and direct method to deduce the classical Blasius’s flat plate equation is considered, as one of the steps to rectify its solution, and shed light on the origin of the conflicting issues involved.

Dual Purpose Design of Small Wind Turbine Blades

Experimental studies of the starting behaviour of small wind turbines have shown that the initial “idling period”, characterised by small rotor acceleration, is usually much longer than the subsequent period of rapid accleration to reach operational rotor speed. Idling obviously reduces the power generation potential of any turbine. The experimental results also imply that most starting torque is generated near the hub, whereas power-producing torque is concentrated near the tip. Therefore this paper considers the computational design of blades for small wind turbines for the dual purposes of (i) efficient power production at rated wind speeds and (ii) rapid starting at smaller wind speeds. Standard blade element theory is used to determine the power coefficient, which is the first objective function to be maximised. A modified blade element method gives the start, whose inverse is the second objective function. During the idling period, the blade angles of attack are relatively large, allowing the lift and drag to be modelled using “generic” flat-plate equations with a small adjustment for the particular aerofoil. No power is extracted from the wind while the blades are accelerated by the aerodynamic torque. Starting is deemed to be complete at a tip speed ratio representing the end of the idling period, which considerably less than the period for power generation. The design is by “differential evolution”. An initial population of blade types is generated randomly and allowed to evolve through random modification and selection of the fittest blades to comprise the next generation. The blade “genes” are the chord and pitch of each blade element; the aerofoil section is the same for all elements and all blades. In multi-objective optimisation, most interest focusses on the members of the population which have at least one objective function greater than every other member, as the final design is usually chosen from these blades. The geometry of the best power extracting blade obtained from the evolutionary optimisation was compared with that arising from standard blade element theory. Good agreement was found for the case of an aerofoil having a pronounced peak in its lift to drag ratio but only a small Reynolds number dependence in the lift coefficient at which the ratio is a maximum. Using an aerofoil with a much flatter lift to drag characteristic, the evolutionary method easily adapted to the significant Reynolds number dependence of the aerofoil characteristics. The effects of blade inertia and the resistive torque of the generator and drive train are also considered. In all cases studied, the best power-producing blade always had relatively poor starting performance, but it was usually possible to choose an alterntive blade with only a small reduction in power output but with a starting time decreased by a factor of around two.

Heat Transfer Downstream of a Leading Edge Separation Bubble

Experiments for flow about a two-dimensional blunt body with a circular leading edge are described. Measurements of the free-stream and boundary-layer velocity distributions are presented and indicate that a small separation “bubble” existed where the leading edge joined the body. In particular, it was found that the laminar leading edge boundary layer separated and reattached shortly downstream as a turbulent boundary layer with a low-momentum thickness Reynolds number. Heat transfer measurements around the body are also presented and show almost an order of magnitude increase across the bubble. Downstream of the bubble, however, the heat transfer could be correlated by a slightly modified turbulent flat plate equation using the separation point as the virtual origin of the heated turbulent boundary layer.

A simplified approach to the large deflection of membranes

The governing differential equations for the large deflection of membranes are obtained from Berger's flat plate equations by letting the plate stiffness tend to zero. Several important problems are discussed, the obtained solutions being in quite good agreement with other approximate analysis.

An experimental investigation of turbulent slot injection at Mach 6

The use of film cooling for protecting a surface exposed to high temperature air at hypersonic speeds is investigated experimentally. The tests were conducted in a Mach 6 contoured axisymmetric nozzle with a streamlined centerbody. The Reynolds number in the test section was in the range of 1-3.6 X 10 6/in.; and a wall to freestream temperature ratio of 0.635. Heat-transfer distributions downstream of the slot were obtained for various mass flow rates and the effect of injection on the velocity, temperature and Mach number profiles was studied. Correlations for the cooling lengths with the blowing rate parameters X = pjUj/peUe for the various coolants-—air, helium, hydrogen, and argon were obtained. Correlations for the heattransfer rates in the form (1 — q/go.») an<l for the effectiveness [e = (Taw. — T0oo)/(T0j — T0oo)] of the adiabatic wall temperatures were obtained as a function of X and x/S (distance from the slot/slot width) for each of the injected coolant gases. The adiabatic wall temperatures are not a basic result of the measurements, but have been inferred from the heat transfer measured, using a local flat plate equation for the same wall temperature. Skin-friction coefficients were also obtained from the measured profiles. Finally the results of this study are compared with available results in the literature.

Local heat-transfer coefficients in a nozzle with high-speed laminarflow.

Abstract : An experimental investigation of the local heattransfer coefficient in the throat region of a nozzle operating under conditions of cold wall, thick laminar boundary-layer, low-density, highspeed flow has been conducted. The experimental results have been compared with several analytic procedures for predicting the heat-transfer coefficient in laminar flow. The simple flat plate equation is shown to underestimate the heat-transfer coefficient, while the method of Cohen and Reshotko predicts coefficients which are too large. The incremental flat plate method of Pasqua and Stevens and a modification of a solution of Beckwith and Cohen exhibit good results downstream of the nozzle throat when reference to the value calculated by the simple flat plate equation at the nozzle exit. The effects of thermal raiation upstream of the nozzle throat are indicated.