A solution of the Navier-Stokes and energy equations illustrating the response of skin friction and temperature of an infinite plate thermometer to fluctuations in the stream velocity

1955 ◽  
Vol 231 (1184) ◽  
pp. 116-130 ◽  
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Skin Friction ◽  
Velocity Fluctuation ◽  
Second Harmonic ◽  
Navier Stokes ◽  
Stream Velocity ◽  
Main Stream ◽  
High Frequencies ◽  
Zero Frequency

An exact solution of the Navier-Stokes equations for incompressible flow is derived under the conditions: (i) the flow is two-dimensional and is bounded by an infinite, plane, porous wall; (ii) the flow is independent of the distance parallel to the wall; (iii) the component of velocity parallel to the wall at a large distance from it fluctuates in time about a constant mean; (iv) the component of velocity normal to the wall is constant. It is found that the skin-friction fluctuations illustrate Lighthill’s (1954) theory of the behaviour of boundary layers subject to fluctuating pressure gradients. The amplitude of the skin-friction fluctuations rises with frequency, while the phase lead of the skin-friction over the main-stream-velocity fluctuation rises from zero at zero frequency to 7r/4 at very high frequencies. The velocity profile in the boundary layer fluctuates, and under certain transient conditions resembles that of a separated boundary layer, that is, a boundary layer with reverse flow close to the wall. With viscous dissipation of kinetic energy taken into account, the corresponding exact solution of the energy equation for an incompressible fluid with constant physical properties is derived under a condition of zero heat transfer between the fluid and the wall—the so-called ‘thermometer’ or ‘kinetic temperature’ problem. Whereas the velocity field consists of a mean flow and a first-harmonic fluctuation, the temperature field contains additionally a second-harmonic fluctuation. It is found that the mean temperature of the wall rises with frequency, and is ultimately proportional to the square root of the frequency. The first-harmonic fluctuation of the wall temperature lags behind the main-stream-velocity fluctuation by an amount which rises from zero at zero frequency to 1/4n at high frequencies, while the phase lag of the second-harmonic rises from zero at zero frequency but drops again to zero at high frequencies. The amplitude of the first-harmonic fluctuation tends to zero at high frequencies, whereas the amplitude of the second-harmonic fluctuation tends to a non-zero limit. Thus the residual temperature fluctuation of the wall at high frequencies has a frequency which is twice that of the fluctuating stream.

Correlated incompressible and compressible boundary layers

1949 ◽  
Vol 200 (1060) ◽  
pp. 84-100 ◽  
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Boundary Layers ◽  
Absolute Temperature ◽  
Stream Velocity ◽  
Main Stream ◽  
Boundary Layer Equations ◽  
Compressible Boundary Layers ◽  
Incompressible Boundary ◽  
Similar Solutions

The boundary-layer equations for a compressible fluid are transformed into those for an incompressible fluid, assuming that the boundary is thermally insulating, that the viscosity is proportional to the absolute temperature, and that the Prandtl number is unity. Various results in the theory of incompressible boundary layers are then taken over into the compressible theory. In particular, the existence of ‘similar’ solutions is proved, and Howarth’s method for retarded flows is applied to determine the point of separation for a uniformly retarded main stream velocity. A comparison with an exact solution is used to show that this method gives a closer approximation than does Pohlhausen’s.

The equation of similar profiles in boundary layer theory with strong blowing

1966 ◽  
Vol 294 (1437) ◽  
pp. 208-234 ◽  
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Main Part ◽  
Similarity Solutions ◽  
Boundary Layer Theory ◽  
Main Stream ◽  
Outer Solution ◽  
Boundary Layer Equations ◽  
Displacement Thickness ◽  
Complicated Form

This paper contains a study of the similarity solutions of the boundary layer equations for the case of strong blowing through a porous surface. The main part of the boundary layer is thick and almost inviscid in these conditions, but there is a thin viscous region where the boundary layer merges into the main stream. The asymptotic solutions appropriate to these two regions are matched to one another when the blowing velocity is large. The skin friction is found from the inner solution, which is independent of the outer solution, but the displacement thickness involves both solutions and is of more complicated form.

Flow Characteristics of Transitional Boundary Layers on an Airfoil in Wakes

2000 ◽  
Vol 122 (3) ◽  
pp. 522-532 ◽  
Author(s):  
H. Lee ◽  
S.-H. Kang
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Skin Friction ◽  
Velocity Fluctuation ◽  
Plate Boundary ◽  
Flow Characteristics ◽  
Turbulence Models ◽  
Transitional Region ◽  
Mean Velocity ◽  
Measured Velocity

Transition characteristics of a boundary layer on a NACA0012 airfoil are investigated by measuring unsteady velocity using hot wire anemometry. The airfoil is installed in the incoming wake generated by an airfoil aligned in tandem with zero angle of attack. Reynolds number based on the airfoil chord varies from 2.0×105 to 6.0×105; distance between two airfoils varies from 0.25 to 1.0 of the chord length. To measure skin friction coefficient identifying the transition onset and completion, an extended wall law is devised to accommodate transitional flows with pressure gradient and nonuniform inflows. Variations of the skin friction are quite similar to that of the flat plate boundary layer in the uniform turbulent inflow of high intensity. Measured velocity profiles are coincident with families generated by the modified wall law in the range up to y+=40. Turbulence intensity of the incoming wake shifts the onset location of transition upstream. The transitional region becomes longer as the airfoils approach one another and the Reynolds number increases. The mean velocity profile gradually varies from a laminar to logarithmic one during the transition. The maximum values of rms velocity fluctuations are located near y+=15-20. A strong positive skewness of velocity fluctuation is observed at the onset of transition and the overall rms level of velocity fluctuation reaches 3.0–3.5 in wall units. The database obtained will be useful in developing and evaluating turbulence models and computational schemes for transitional boundary layer. [S0098-2202(00)01603-5]

On the generation of sound by turbulent boundary layer flow over a rough wall

1984 ◽  
Vol 395 (1809) ◽  
pp. 247-263 ◽  
Keyword(s):  
Boundary Layer ◽  
Near Field ◽  
Spectral Characteristics ◽  
Rigid Wall ◽  
Control Surface ◽  
Aerodynamic Noise ◽  
Stream Velocity ◽  
Main Stream ◽  
Wall Roughness ◽  
Boundary Layer Turbulence

The production of sound by scattering of the near field of low Mach number boundary-layer turbulence by a rough, rigid wall is examined on the basis of Lighthill’s theory ( Proc. R. Soc. Lond . A 211, 564 (1952)) of aerodynamic noise. The radiation is expressed in terms of the turbulence pressure spectrum on a control surface that is parallel to the mean plane of the wall and at a stand-off distance equal to the height of the wall roughness elements, the surface irregularities being modelled by a distribution of hemispherical bosses on an otherwise plane wall. The intensity of the sound produced by unit area of the wall varies as the sixth power of the main stream velocity and, for given wall roughness, increases as the boundary-layer thickness decreases. These conclusions are in accord with experimental observations reported by Hersh { AIAA paper no. 83-0786) of the generation of high frequency sound by turbulent flow from sand-roughened pipes, and it is shown how, for moderately rough pipes, the theory reproduces the spectral characteristics of Hersh’s data.

The Effects of Small Changes of Prandtl Number and the Viscosity-Temperature Index on Skin Friction

1964 ◽  
Vol 15 (4) ◽  
pp. 392-406 ◽  
Author(s):  
A. D. Young
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Pressure Gradient ◽  
Skin Friction ◽  
Laminar Boundary Layer ◽  
Wall Temperature ◽  
External Pressure ◽  
Main Stream ◽  
Pressure Gradients ◽  
Temperature Index

SummaryThe analytic simplifications in boundary-layer analysis that result from the assumptions that the Prandtl number σ and the viscosity-temperature index ω are unity make it desirable to be able to assess the effects of the departures of the actual values of these parameters from unity. In this paper only the effects on skin friction are considered. Formulae of acceptable validity and wide application are first used to produce generalised curves for these effects for given main-stream Mach numbers and wall temperature conditions for the case of zero external pressure gradient for both laminar and turbulent boundary layers (Figs. 1 and 2).A number of calculated results for the laminar boundary layer with favourable and adverse pressure gradients is then analysed (Figs. 3, 4 and 5) and it is shown that these results are consistent with the assumption that, for a given wall temperature, the effects of small changes of σ and ω on skin friction are independent of the external gradient, so that the appropriate curves of Figs. 1 and 2 apply. Where the change of a- is associated with a change of wall temperature (e.g. if the heat transfer is specified as zero) then the interaction between pressure gradient and this temperature change can be significant in its effects on skin friction for the laminar boundary layer and can only be assessed if the effects of changes of wall temperature with constant σ and ω have been separately determined for the pressure distribution considered. It is inferred that in all cases, except with large adverse pressure gradients and imminent separation, the effects of changes of ω and σ for the turbulent boundary layer are reliably predicted by the zero pressure gradient curves of Figs. 1 and 2 and the effect of any associated change of wall temperature can then be reliably inferred from the zero pressure gradient formula (equation (15)) in the absence of more specific calculations covering a range of wall temperatures.

The structure of two-dimensional separation

1990 ◽  
Vol 220 ◽  
pp. 397-411 ◽  
Author(s):  
Laura L. Pauley ◽  
Parviz Moin ◽  
William C. Reynolds
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Adverse Pressure Gradient ◽  
Navier Stokes ◽  
Stream Velocity ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Shedding Frequency

The separation of a two-dimensional laminar boundary layer under the influence of a suddenly imposed external adverse pressure gradient was studied by time-accurate numerical solutions of the Navier–Stokes equations. It was found that a strong adverse pressure gradient created periodic vortex shedding from the separation. The general features of the time-averaged results were similar to experimental results for laminar separation bubbles. Comparisons were made with the ‘steady’ separation experiments of Gaster (1966). It was found that his ‘bursting’ occurs under the same conditions as our periodic shedding, suggesting that bursting is actually periodic shedding which has been time-averaged. The Strouhal number based on the shedding frequency, local free-stream velocity, and boundary-layer momentum thickness at separation was independent of the Reynolds number and the pressure gradient. A criterion for onset of shedding was established. The shedding frequency was the same as that predicted for the most amplified linear inviscid instability of the separated shear layer.

Direct numerical simulation of spatially developing turbulent boundary layers with uniform blowing or suction

2011 ◽  
Vol 681 ◽  
pp. 154-172 ◽  
Author(s):  
YUKINORI KAMETANI ◽  
KOJI FUKAGATA
Keyword(s):  
Numerical Simulation ◽  
Boundary Layer ◽  
Direct Numerical Simulation ◽  
Drag Reduction ◽  
Skin Friction ◽  
Reduction Factor ◽  
Stream Velocity ◽  
Normal Velocity ◽  
Friction Drag ◽  
Local Skin

Direct numerical simulation (DNS) of spatially developing turbulent boundary layer with uniform blowing (UB) or uniform suction (US) is performed aiming at skin friction drag reduction. The Reynolds number based on the free stream velocity and the 99% boundary layer thickness at the inlet is set to be 3000. A constant wall-normal velocity is applied on the wall in the range, −0.01U∞ ≤ Vctr ≤ 0.01U∞. The DNS results show that UB reduces the skin friction drag, while US increases it. The turbulent fluctuations exhibit the opposite trend: UB enhances the turbulence, while US suppresses it. Dynamical decomposition of the local skin friction coefficient cf using the identity equation (FIK identity) (Fukagata, Iwamoto & Kasagi, Phys. Fluids, vol. 14, 2002, pp. L73–L76) reveals that the mean convection term in UB case works as a strong drag reduction factor, while that in US case works as a strong drag augmentation factor: in both cases, the contribution of mean convection on the friction drag overwhelms the turbulent contribution. It is also found that the control efficiency of UB is much higher than that of the advanced active control methods proposed for channel flows.

Skin Friction and Heat Transfer for Laminar Boundary-Layer Flow With Variable Properties and Variable Free-Stream Velocity

1953 ◽  
Vol 20 (3) ◽  
pp. 415-421
Author(s):  
S. Levy ◽  
R. A. Seban
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Skin Friction ◽  
Laminar Boundary Layer ◽  
Boundary Layer Flow ◽  
Free Stream ◽  
Free Stream Velocity ◽  
Stream Velocity ◽  
Layer Flow

Abstract Numerical solutions of the momentum and energy equations are presented for particular types of laminar boundary-layer flow analogous to the Hartree “wedge flows.” Variation of the viscosity and of the thermal conductivity is considered under the circumstances of no dissipation, favorable pressure gradient, and the product of conductivity and density a constant. The solution is based on approximate representations of the velocity and temperature profiles in the boundary layer and these are of such character that the labor of calculation is minimized and the accuracy of the results preserved. The differential equations are reduced to two algebraic equations which rapidly yield the skin friction and the heat transfer in terms of the wall to free-stream temperature ratio for the desired value of Prandtl number. Numerical results are given for a range of wedge flows with gases of Prandtl number 0.70 and 1.0. These results reveal that when the free-stream velocity is variable the temperature difference between the wall and the free stream exerts a substantial effect on the velocity distribution in the boundary layer and on the skin-friction coefficient. Alternatively, the heat-transfer coefficient is not affected radically. A calculation method is presented for the determination of the heat transfer and skin friction for a flow with an arbitrary variation of velocity over an isothermal surface. This method utilizes the results of the present analysis for the variable property wedge flows.

Contributions to the theory of heat transfer through a laminar boundary layer

1950 ◽  
Vol 202 (1070) ◽  
pp. 359-377 ◽  
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Boundary Layer ◽  
Temperature Distribution ◽  
Skin Friction ◽  
Heat Transfer Rate ◽  
Wall Temperature ◽  
Transfer Rate ◽  
Arbitrary Distribution ◽  
Main Stream

An approximation to the heat transfer rate across a laminar incompressible boundary layer, for arbitrary distribution of main stream velocity and of wall temperature, is obtained by using the energy equation in von Mises’s form, and approximating the coefficients in a manner which is most closely correct near the surface. The heat transfer rate to a portion of surface of length l (measured downstream from the start of the boundary layer) and unit breadth is given as -½ k /(⅓)! (3σρ/μ 2 ) ⅓ ∫ l 0 (∫ l x √{ T ( z )} dz ) ⅔ dT 0 ( x ), where k is the thermal conductivity of the fluid, σ its Prandtl number, ρ its density, μ its viscosity, T ( x ) is the skin friction, and T 0 ( x ) the excess of wall temperature over main stream temperature. A critical appraisement of the formula (§3) indicates that it should be very accurate for large σ, but that for σ of order 0.7 (i. e. for most gases) the constant ½3 ⅓ /(⅓) ! = 0.807 should be replaced by 0.73, when the error should not exceed 8 % for the laminar layers that occur in practical aerodynamics. This yields a formula Nu = 0.52σ ⅓ ( R √ C f ) ⅔ for Nusselt number in terms of the Reynolds number R and the mean square root of the skin friction coefficient C f , in the case of uniform wall temperature. However, for the boundary layer with uniform main stream, the original formula is accurate to within 3% even for σ = 0.7. By known transformations an expression is deduced for heat transfer to a surface, with arbitrary temperature distribution along it, and with a uniform stream outside it at arbitrary Mach number (equation (42)). From this, the temperature distribution along such a surface is deduced (§ 4) in the case (of importance at high Mach numbers) when heat transfer to it is balanced entirely by radiation from it. This calculation, which includes the solution of a non-linear integral equation, gives higher temperatures near the nose, and lower ones farther back (figure 2), than are found from a theory which assumes the wall temperature uniform and averages the heat transfer balance. This effect will be considerably mitigated for bodies of high thermal conductivity; the author is not in a position to say whether or not it will be appreciable for metal projectiles. But for stony meteorites at a certain stage of their flight through the atmosphere it indicates that melting at the nose and re-solidification farther back may occur, for which the shape and constitution of a few of them affords evidence. An appendix shows how the method for approximating and solving von Mises’s equation could be used to determine the skin friction as well as heat transfer rate, but this line seems to have no advantage over established approximate methods.

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