Fluctuating Pressure Gradients Research Articles

A pdf modeling study of self-similar turbulent free shear flows

A modeled transport equation for the joint probability density function (pdf) of the velocities and a scalar has been solved numerically for four self-similar turbulent free shear flows: the plane mixing layer, the plane wake, the plane jet, and the axisymmetric jet. In the pdf equation, convection is treated exactly but the effects of viscosity, molecular diffusion, and the fluctuating pressure gradient have to be modeled. Five different models are evaluated; four of these are based on the Langevin equation for the fluid particle velocities, and the fifth is a particle pair interaction model. In each case, a stochastic mixing model represents the effects of molecular diffusion, and conditional modeling is included to account for intermittency. The resulting modeled pdf equation is solved by a Monte Carlo method. Calculated spreading rates and profiles of the mean velocity and Reynolds stresses show generally good agreement with experimental data for three of the five velocity models. However, discrepancies between the calculated and measured intermittency factor profiles, in particular for the plane mixing layer, reveal weaknesses in the conditional modeling.

Vortex Simulation of Unsteady Wrinkled Laminar Flames

A discrete vortex dynamics time-dependent simulation of a premixed wrinkled laminar flame shows that the computed turbulent shear stress in the flame brush region based on unconditioned velocities is substantial. By comparison, the computed shear stresses in the burnt and in the unburnt flow are negligible. Thus, it is the intermittent flame motion that gives the appearance of counter-gradient fluxes in a turbulence model based on unconditioned velocities. The flame is considered to be of zero thickness and the flame location is described in terms of arc length from the flame holder. Two-dimensional V-shaped and bunsen burner flames have been simulated in planar geometry. A flame speed relation is used which depends on flame curvature and in some calculations on flame stretch. Discrete volume sources are distributed along the flame location to represent the volume expansion at constant pressure combustion in this open system. The estimated vorticity production due to fluctuating pressure gradients combined with an assumed flame density gradient is small compared to the freestream turbulence level of five percent, and hence was neglected in this simulation. At fixed locations in the middle of the flame brush, the passage time distribution indicates the geometrical nature of the flame sheet in that the cusps that point towards the burnt fluid produce an increased probability at short durations for unburned fluid in comparison with the burnt passage times.

Structure and blow-off mechanism of rod-stabilized premixed flame

In order to determine the key feature in the blow-off and to reexamine the combustion mechanism, an experimental investigation was performed on propaneair flames stabilized by cylindrical rod bluff-bodies when the number and diameter of rods were changed under a constant blockage ratio of 0.4. Measurements were made of pressure gradient fluctuations along the rod surface and the rms values of ion current on the wake axis. The development of a flame generated at the attachment point was also pursued by using high-speed schlieren photography and ionization probes. The results showed that the instantaneous flame structure is composed of a series of small scale eddy-flames and large scale lump-flames containing a small definite number of eddy-flames. The attached flame causes a locally intense fluctuation of the surface pressure gradient at the attachment point probably due to flame-flow interaction. The lump-flames originate in the strongly stretched eddy-flames because of the maximum velocity gradient across a boundary layer at that point, and the surface pressure gradient fluctuation governs the process of lump-flame formation. It is also shown that the recirculation zone length varies with the range of L R d = 1.5–6.0 , which agrees well with that of the lump-flame formation point Lcd = 1.4–5.8. From these results a local extinction of the excessively stretched weak eddy-flames at the end of the recirculation zone was found to trigger the blow-off of the rod-stabilized flame. This enabled a qualitative modeling of the mechanisms of the lump-flame formation and flame stabilization behind the rods.

Fluid dynamics of a flow excited resonance, Part II: Flow acoustic interaction

This is the second of two companion papers in which the physics and detailed fluid dynamics of a flow excited resonance are examined. The approach is rather different from those previously used, in which stability theory has been applied to small wavelike disturbances in a linearly unstable shear layer, with an equivalent source driving the sound field which provides the feedback. In the approach used here, the physics of the flow acoustic interaction is explained in terms of the detailed momentum and energy exchanges occurring inside the fluid. Gross properties of the flow and resonance are described in terms of the parameters necessary to determine the behaviour of the feedback system. In this second paper it is shown that two relatively distinct momentum balances can be considered in the resonator neck region. One can be identified with the vortically induced pressure and velocity fluctuations and the other with the reciprocating potential flow. The fluctuating Coriolis force caused by the interaction of the potential and vortical flows is shown to be the only term in the linearized momentum equation which is not directly balanced by a fluctuating pressure gradient. This force provides the mechanism for the exchange of the mean energies associated with the mean and fluctuating momenta, respectively. A source and sink of energy are identified in which mean energy associated with fluctuating momentum is extracted from and returned to the mean flow, respectively. The imbalance between the source and sink is responsible for both the radiated acoustic power and the power carried away by the vortices as they convect downstream. This radiated acoustic power and vortically convected power, and the source and sink powers, are all of the same order of magnitude. With the vortex shedding and reciprocating potential flow “phase locked” the amplitude of the steady state oscillations is determined by the condition that the net power produced in the resonator neck (the source power less the sink power) is equal to the sum of the radiated acoustic power and that carried by the vortices.

Pressure-Velocity Correlation in a Reactive Turbulent Flow

Analysis is made of the often-neglected correlation between velocity and pressure gradient fluctuations which occurs in the turbulent kinetic energy equation (TKEE). Restricting attention to this correlation for normal turbulent flames in premixed gases using high activation energy asymptotics, a model is generated for the correlation. Splitting the velocity field into solenoidal and irrotational components, it is shown that the acoustic field can dominate the vortical field under several circumstances, in which case the TKEE is not a turbulence equation but an acoustic equation. Under conditions where the vortical field is dominant the pressure-velocity correlation is shown to be essential to the TKE balance.

Flow enhancement mechanisms of a pulsating flow of non-Newtonian liquids

The pipe flow of non-Newtonian liquids under a fluctuating pressure gradient is considered. Adopting a generalized Maxwell model with full relaxation spectrum and considering only weak-sense stationary pressure-gradient noises, it is shown that there are two mechanisms involved in the flow enhancement: an inelastic and a dynamic mechanism. Both depend on the shear-thinning properties of the liquid in steady and oscillatory flows. For one-frequency pressuregradient noises, the flow enhancement increases with increasing frequency of fluctuation if the modulus of the complex viscosity is a decreasing function of the frequency. Since almost all polymeric liquids are shear-thinning in oscillatory shear flows, this investigation serves as a possible explanation for the data collected byBarnes, Townsend andWalters (1971). This, however, does not necessarily ensure that the constitutive model adopted in this study is free of defect in a general flow field.

On pulsating flow of polymeric fluids

The flow problem of an elastico-viscous fluid in a straight circular pipe under the influence of a fluctuating pressure gradient is considered. Adopting a simple, generalized Maxwell model, it is shown that the percentage increase in the mean flow rate rises with increasing frequency of fluctuation. The value of the mean pressure gradient at which the flow rate enhancement attains its maximum value is determined entirely by inelastic calculations.

Three-Dimensional Characteristics of Diurnally Varying Boundary-Layer Flows

Abstract Numerical models of three-dimensional, diurnally varying, boundary-layer flows are integrated to study the effect of fluctuating pressure gradients and eddy stresses within different circulation systems. The computational problem is reduced by expanding the horizontal dependence of solutions into Taylor series truncated at the first two terms. Within this simplification, sufficient generality is retained to reproduce axially symmetric similarity solutions and solutions of the nonlinear balance equation. For the time-dependent cases, substantial deviations from linearized (horizontally uniform) theory are predicted. Diurnally periodic pressure gradient and eddy stress oscillations cause greatly differing responses for various circulation systems. The magnitude of the balanced vorticity and the nature of the local deformation field have great bearing on the development of boundary-layer jets and secondary vertical circulations.

Acoustic radiation from a turbulent fluid containing foreign bodies

The problem of the acoustic radiation from a turbulent fluid containing foreign bodies of arbitrary shape and arbitrary composition is solved formally by the method of Green functions. Particular attention is given to the radiation from the surface of these bodies. A practical advantage of the method is that provided an appropriate Green function can be found, either exactly or approximately, then knowledge of the values on the surface of the fluctuations in only one scalar variable is needed to permit estimation of the radiation from the surface. This variable may be either the pressure, the normal pressure gradient, the density, or the normal density gradient. The pressure fluctuations at the surface, in particular, are relatively easy to measure. It is shown that if fluctuations in the fluid are locally isentropic the volume source distribution of the pressure fluctuations is quadrupole. A proof is given of the proposition that when arbitrary obstacles are immersed in a fluid all dipole radiation must come from surface source distributions on these obstacles. For rigid bodies these distributions represent physically the reaction by the obstacles to the stresses imposed upon them by the fluid. It is pro ved that if the density fluctuations or the normal density gradient fluctuations on these surfaces vanish then there is no dipole radiation. The same result is true for pressure and pressure gradient fluctuations within the limits of validity of the assumption of local isentropy. A brief description is given, together with references to more detailed accounts, of some of the principal features of the behaviour of Green functions which may be useful in practical estimates of aerodynamic surface sound. As a representative example of acoustic radiation from a turbulent boundary layer, the total acoustic power radiated by a turbulent boundary layer on an infinite rigid plane is estimated, using the limited available experimental data on wall pressure fluctuations. For low and moderate subsonic speeds the power radiated per unit wall area covered by the turbulent boundary layer is Kρ 0 a 3 0 M 6 0 , where M 0 is the free stream Mach number, ρ 0 is the density and a 0 the speed of sound in the undisturbed fluid, and the dimensionless parameter K is approximately a constant of order of magnitude 10 -5 .

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

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.

The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid

1. Taylor has shown (1) that two characteristic lengthsλ and λη may be defined for turbulent fluid motion. The length λ, which is connected with the dissipation of energy, is, for isotropic turbulence, given bywhere is the mean rate of dissipation of energy per unit volume and represents the mean square value of any component of velocity. The length λη can be defined in terms of thuswhere For isotropic turbulence Taylor assumed thatwhere B is a constant. Since the turbulence is isotropic,and so, from (1), (2), (3) and (4) we have

The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid

In the general systems of vortices represented by (1) the mean variation in pressureiswhereKis a number which varies between 1 and √2. When the vortices are confined to cubical partitions, the case most nearly analogous to that of free turbulence,K= 1·06, so that the conjecture which I made some years ago, thatwould be equal tois probably nearly correct.