Re-Entrant Corner for a White-Metzner Fluid

The effects of inertia on the upstream-growing boundary layer over a finite horizontal flat plate of length b moving uniformly with speed U0 in a linearly stratified \[ [(dp/dy)_{-\infty}- = - \rho_0\beta)], \] viscous, non-diffusive fluid under the Boussinesq approximation are studied. The nonlinear inertia terms are linearized by the Oseen approximation, but no boundary-layer approximation is required. The flow is governed by two parameters, namely the internal Froude number Fr[= U0/(βgb2)½] and a parameter L3[= βgb3/U0ν], where $L^{\frac{3}{4}}$ is proportional to the ratio of boundary-layer thickness to plate length for the case Fr = 0. Large values of L3 and Fr2 = 0 correspond to the case of an upstream boundary layer. By increasing the Froude number gradually, a transition occurs from an upstream boundary layer accompanied by an upstream wake to a downstream boundary layer with a downstream wake. The upstream boundary layer and wake are characterized by a balance of viscous and buoyancy forces, whereas the downstream boundary layer and wake are characterized by a balance of inertia and viscous forces. In the so-called critical-boundary-layer case, Fr4L3 = O(1), inertia, viscous and buoyancy forces are all important and this boundary layer is accompanied by both upstream and downstream wakes. Complete transition occurs when Fr4L3 increases from 10·0 to 1000·0. The drag on the plate is also calculated.

Spatial coherence of low-frequency unsteadiness associated with a normal shock wave

Experimental study is performed to understand and quantify the wall and eccentric retardation effects on spheres settling in shear thinning and shear-thinning viscoelastic fluids over a wide range of diameter ratios (0.02 < λ < 0.9). The four-parameter Carreau viscosity equation has been chosen to represent the apparent viscosity-shear rate of polyacrylamide solutions. Two new wall factor corrections are presented with excellent agreement compared to experimental data.The terminal settling velocity of a sphere in bounded fluid is significantly reduced by the presence of confining boundaries, named wall retardation effect that decreases due to the shear-thinning behavior of power law fluids, which is weaken further by the elastic effect of viscoelastic fluids. The wall factors of spheres settling in viscoelastic fluids increase at low ξ up to 50, followed by a horizontal confidence region (0.7 ≤ f ≤ 1) at high ξ. In this region, the wall factor is mainly dominated by fluids’ elasticity, which is more distinguished for small spheres. As the settling spheres approach to the wall (b/R → 1), the neighboring wall exert more intensive retardation that reduce the terminal settling velocity greatly when b/R > 0.6 in pure shear-thinning fluids, and the extra retardation effect of nearby wall increases at high concentration due to the enhanced non-Newtonian property. In contrast, the eccentric effect on settling velocity in viscoelastic fluids is cut down greatly by the fluid's elasticity, which is negligible.

The K-type and H-type transitions of a natural convection boundary layer of a fluid of Prandtl number 7 adjacent to an isothermally heated vertical surface are investigated by means of three-dimensional direct numerical simulation (DNS). These two types of transitions refer to different flow features at the transitional stage from laminar to turbulence caused by two different types of perturbations. To excite the K-type transition, superimposed Tollmien–Schlichting (TS) and oblique waves of the same frequency are introduced into the boundary layer. It is found that a three-layer longitudinal vortex structure is present in the boundary layer undergoing the K-type transition. The typical aligned $\wedge$-shaped vortices characterizing the K-type transition are observed for the first time in pure natural convection boundary layers. For exciting the H-type transition, superimposed TS and oblique waves of different frequencies, with the frequency of the oblique waves being half of the frequency of the TS waves, are introduced into the boundary layer. Unlike the three-layer longitudinal vortex structure observed in the K-type transition, a double-layer longitudinal vortex structure is observed in the boundary layer undergoing the H-type transition. The successively staggered $\wedge$-shaped vortices characterizing the H-type transition are also observed in the downstream boundary layer. The staggered pattern of $\wedge$-shaped vortices is considered to be caused by temporal modulation of the TS and oblique waves. Interestingly the flow structures of both the K-type and H-type transitions observed in the natural convection boundary layer are qualitatively similar to those observed in Blasius boundary layers. However, an analysis of turbulence energy production suggests that the turbulence energy production by buoyancy rather than Reynolds stresses dominates the K-type and H-type transitions. In contrast, the turbulence energy production by Reynolds stresses is the only factor contributing to the transition in Blasius boundary layers.

A perturbation solution is obtained for the flow of a viscoelastic fluid through a channel whose radius varies slowly in the direction of the flow. The perturbation expansion is carried out in a small parameter used by Manton for a Newtonian flow which prescribes how slowly the radius of the channel varies. An integral rheological model with no shear thinning is used for the viscoelastic fluid. The solution is applicable to flows of low Reynolds number and low Weissenberg number. The normal and shear stresses at walls of the channel are also presented as well as the stream function. The solution is illustrated for two examples : a locally constricted channel and a converging channel, and it is generally discussed how the difference in behavior between viscoelastic and Newtonian fluids depends on the shapes of channels.

This note gives a simple‐minded approach to the two‐dimensional boundary layer equations. The pressure is eliminated from the equations of motion and the resulting equation is simplified by assuming that certain derivatives in the direction of the boundary are small compared with those at right angles to it. The simplified equation is then integrated to give a single boundary layer equation which, together with the stress rate of strain law and the continuity equation, is sufficient (in theory at least) to predict the flow. The boundary layer equation as given does not depend on a particular form for the stress rate of strain law and could possibly form the basis for a non‐Newtonian investigation. The viscous boundary layer is given as a special case.

Momentum boundary layer of viscoelastic second grade fluid taken into consideration with the effect of uniform transverse magnetic field and normal electric field also included in the analysis. Thermal boundary layer formed over the non-isothermal boundary wall, with viscous dissipation and Ohmic dissipation due to transverse magnetic field and electric field has been investigated. The highly non-linear momentum boundary layer equation and thermal boundary layer equation are converted into similarity equations and then solved numerically by employing fifth order Runge-Kutta-Fehlberg method with shooting.

The famous Toms effect (1948) consists of a substantial increase of the critical Reynolds number when a small amount of soluble polymer is introduced into water. The most noticeable influence of polymer additives is manifested in the boundary layer near solid surfaces. The task includes the ratio of two characteristic length scales, one of which is the Prandtl scale, and the other is defined as the square root of the normalized coefficient of relaxation viscosity (Frolovskaya and Pukhnachev, 2018) and does not depend on the characteristics of the motion. In the limit case, when the ratio of these two scales tends to zero, the equations of the boundary layer are exactly integrated. One of the goals of the present paper is group analysis of the boundary layer equations in two mathematical models of the flow of aqueous polymer solutions: the second grade fluid (Rivlin and Ericksen, 1955) and the Pavlovskii model (1971). The equations of the plane non-stationary boundary layer in the Pavlovskii model are studied in more details. The equations contain an arbitrary function depending on the longitudinal coordinate and time. This function sets the pressure gradient of the external flow. The problem of group classification with respect to this function is analyzed. All functions for which there is an extension of the kernels of admitted Lie groups are found. Among the invariant solutions of the new model of the boundary layer, a special place is taken by the solution of the stationary problem of flow around a rectilinear plate.

Stagnation-point flow of a viscoelastic fluid towards a stretching surface

An analysis is made on a steady flow of an incompressible visco-elastic fluid (of small memory) past a porous plate subject to uniform suction or blowing. The fluid and the plate rotate in unison with uniform angular velocity about an axis perpendicular to the plate. It is shown that for a visco-elastic fluid of small memory, a boundary layer is formed for uniform suction or blowing at the plate. It is found that the primary velocity increases whereas the secondary velocity increases near the plate and decreases away from the plate with an increase in the elastic parameter. The secondary flow shows flow reversal for a large elastic parameter. It is found that both primary and secondary velocities increase near the plate and decrease away from the plate with an increase in the rotation parameter. It is also found that the shear stress due to primary flow increases with an increase in either the elastic parameter or rotation parameter. On the other hand, the shear stress due to secondary flow decreases with an increase in the elastic parameter while it increases with an increase in the rotation parameter. The temperature distribution in the boundary layer is also determined. It is shown that elasticity leads to an increase in heat transfer at the plate.

This work presents an experimental investigation for effects of the spanwise spacing of small-jets placed in the approaching boundary layer on cavity tone. The sound and velocity fields of a flow around a cavity were measured with a low noise wind tunnel. The freestream velocity was changed from U_0 = 10-45 m/s. The depth-to-length ratio of the cavity was D/L = 0.5. The upstream boundary layer is laminar, and the boundary layer thickness is δ_<99%>/L = 0.08 at U_0 = 43 m/s. The effects of the jets from the wall in the upstream boundary layer with various spanwise spacing p/L = 0.1, 0.25, 0.6, and 1.25 on the cavity tone were investigated. The jet velocity at the central axis of jet was changed from U_j = 2.2-4.0 m/s. The results show the cavity tone can be reduced by control of blowing jets. The control effect on the cavity tone depends on the spanwise pitch and velocity of blowing jets. In this experiment, the most effective pitch of jets for noise reduction was p/L = 0.25. The reduction level increases as the jet velocity becomes higher. The noise reduction effect becomes sufficiently large when the jet velocity is higher than 3 m/s. The three-dimensional spanwise velocity field was induced by blowing jets. As a result, the spanwise coherence of the vortices inducing intense sound became weak, and the radiating sound became weak.

Nitrogen oxides have been shown to catalyze the destruction of stratospheric ozone. Previous estimates of nitric oxide emissions from high‐altitude aircraft have taken into account engine emissions only. This study was undertaken to determine whether nitric oxide production in the hot viscous boundary layer surrounding the skin of a hypersonic aircraft can significantly increase estimates of total NO emissions. The described model approximates the viscous boundary layer as a series of finite stream tubes with time‐dependent and temperature‐dependent chemical kinetics and mass flow rates. Nitric oxide equilibrium mole fraction for air peaks at ≅ 3700 K. At hypersonic speeds the viscous effects near the aircraft skin will induce these very high temperatures. Along the skin surface of a 60 m craft, the NO mole fraction can be locally as high as 0.044. Nitric oxide output from this thin boundary layer becomes significant when integrated over the entire trailing edge of the aircraft. This model predicts that nitric oxide production in the boundary layer should be taken into account at speeds above Mach 8. Above Mach 8, boundary layer volume and temperature increase rapidly. At stratospheric speeds approaching Mach 16, the nitric oxide production in the boundary layer increases to the point where it roughly equals the nitric oxide output from the engines. Above Mach 16, the boundary layer produces the majority of the NO emissions. This additional source of nitric oxide will play an ever‐increasing role as aircraft fly at higher speeds and altitudes. Significant errors in total NO emission estimates will result if the viscous boundary layer is not taken into account at speeds above Mach 8.

This paper focuses on investigating the influence of a back chamber on the flow stability in an open-cavity model. First, the global stability analysis employed in the study is introduced and validated, followed by a description of the open-cavity configuration equipped with a back chamber. Base flow computations reveal that the introduction of a back chamber leads to the formation of three vortex structures: a Primary Eddy, an UpStream Eddy, and a DownStream Eddy. The global stability analysis is employed to conduct investigations into the effects of two- and three-dimensional disturbances, Reynolds numbers, and the corresponding direct global eigenmodes. The study further examines the inviscid centrifugal instability mechanism responsible for triggering three-dimensional flow instability in the field. Furthermore, this study tries to relate the studied model to the compressor physics. On one hand, axial and radial momentum source terms are introduced into the flow field to investigate their effects on the downstream boundary layer. On the other hand, unsteady numerical simulations are conducted to examine the impact of upstream-propagating disturbances, caused by a subsonic compressor, on the incoming flow boundary layer. The results indicate that the influence of both models on the downstream boundary layer is essentially consistent. The introduction of a back chamber provides additional volume to dissipate the energy of the upstream-propagating disturbances.

The development of a boundary layer over a flat plate placed in a square pipe was studied in pulp suspensions at concentrations of 0.5, 1 and 1.8% at flow velocities of 0.24-0.96 m/s. Laser Doppler Anemometry (LDA) was used to measure velocities in the boundary layer from y=0-3 mm at x=0, 5, 10, 20 and 30 mm, where x=0 mm was the position where the flow met the plate. During the measurement time the pressure drop over the plate was also measured and it was concluded that the flow was in the rolling friction regime. The results of boundary layer thickness show that an increase in velocity or a decrease in concentration gave a thicker boundary layer in contrast to a laminar Newtonian boundary layer. In a power law fluid, boundary layer thickness decreases with an increase in shear thinning or velocity. Furthermore, in the pulp suspensions, a boundary layer with a constant thickness was reached 20 mm from the plate edge in contrast to a Newtonian and a shear thinning boundary layer which continued to increase along the plate. The difference in boundary layer development is believed to depend on the two-phase effects in pulp suspensions.

A numerical analysis is made of the dynamic boundary layer and the thermal boundary layer at a sendinfinitely large plate longitudinally streamlined by a viscoelastic fluid.