Brian J. Cantwell

A general classification of three‐dimensional flow fields

The geometry of solution trajectories for three first‐order coupled linear differential equations can be related and classified using three matrix invariants. This provides a generalized approach to the classification of elementary three‐dimensional flow patterns defined by instantaneous streamlines for flow at and away from no‐slip boundaries for both compressible and incompressible flow. Although the attention of this paper is on the velocity field and its associated deformation tensor, the results are valid for any smooth three‐dimensional vector field. For example, there may be situations where it is appropriate to work in terms of the vorticity field or pressure gradient field. In any case, it is expected that the results presented here will be of use in the interpretation of complex flow field data.

An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder

This paper describes an experimental investigation of transport processes in the near wake of a circular cylinder at a Reynolds number of 140000. The flow in the first eight diameters of the wake was measured using X-array hot-wire probes mounted on a pair of whirling arms. This flying-hot-wire technique increases the relative velocity component along the probe axis and thus decreases the relative flow angle to usable values in regions where fluctuations in flow velocity and direction are large. One valuable fringe benefit of the technique is that rotation of the arms in a uniform flow applies a wide range of relative flow angles to the X-arrays, making them inherently self-calibrating in pitch. An analog circuit was used to generate an intermittency signal, and a fast surface-pressure sensor was used to generate a phase signal synchronized with the vortex-shedding process. The phase signal allowed sorting of the velocity data into 16 populations, each having essentially constant phase. An ensemble average for each population yielded a sequence of pictures of the instantaneous mean flow field, with the vortices frozen as they would be in a photograph. In addition to globally averaged data for velocity and stress, the measurements yield non-steady mean data (in the sense of an average a t constant phase) for velocity, intermittency, vorticity, stress and turbulent-energy production as a function of phase for the first eight diameters of the near wake. The stresses were resolved into a contribution from the periodic motion and a contribution from the random motion at constant phase. The two contributions are found to have comparable amplitudes but quite different geometries, and the time average of their sum (the conventional global Reynolds stress) therefore has a quite-complex structure. The non-steady mean-vorticity field is obtained with good resolution as the curl of the non-steady mean-velocity field. Less than half of the shed circulation appears in the vortices, and there is a slow decay of this circulation for each shed vortex as it moves downstream. In the discussion, considerable emphasis is put on the topology of the non-steady mean flow, which emerges as a pattern of centres and saddles in a frame of reference moving with the eddies. The kinematics of the vortex-formation process are described in terms of the formation and evolution of saddle points between vortices in the first few diameters of the near wake. One important conclusion is that a substantial part of the turbulence production is concentrated near the saddles and that the mechanism of turbulence production is probably vortex stretching at intermediate scales. Entrainment is also found to be closely associated with saddles and to be concentrated near the upstream-facing interface of each vortex.

Introduction to Symmetry Analysis

Preface 1. Introduction to symmetry 2. Dimensional analysis 3. Systems of ODEs, first order PDEs, state-space analysis 4. Classical dynamics 5. Introduction to one-parameter Lie groups 6. First order ordinary differential equations 7. Differential functions and notation 8. Ordinary differential equations 9. Partial differential equations 10. Laminar boundary layers 11. Incompressible flow 12. Compressible flow 13. Similarity rules for turbulent shear flows 14. Lie-Backlund transformations 15. Invariance condition for integrals, variational symmetries 16. Backlund transformations and non-local groups Appendix 1. Review of calculus and the theory of contact Appendix 2. Invariance of the contact conditions under Lie point transformation groups Appendix 3. Infinite-order structure of Lie-Backlund transformations Appendix 4. Symmetry analysis software.

Structure and entrainment in the plane of symmetry of a turbulent spot

Laser-Doppler velocity measurements in water are reported for the flow in the plane of symmetry of a turbulent spot. The unsteady mean flow, defined as an ensemble average, is fitted to a conical growth law by using data at three streamwise stations to determine the virtual origin in x and t. The two-dimensional unsteady stream function is expressed as ψ=U^2_∞tg(ξ,η) in conical similarity co-ordinates ζ = x/U_∞t and η = y/U_∞t. In these co-ordinates, the equations for the unsteady particle displacements reduce to an autonomous system. This system is integrated graphically to obtain particle trajectories in invariant form. Strong entrainment is found to occur along the outer part of the rear interface and also in front of the spot near the wall. The outer part of the forward interface is passive. In terms of particle trajectories in conical co-ordinates, the main vortex in the spot appears as a stable focus with celerity 0·77U_∞. A second stable focus with celerity 0·64U_∞ also appears near the wall at the rear of the spot. Some results obtained by flow visualization with a dense, nearly opaque suspension of aluminium flakes are also reported. Photographs of the sublayer flow viewed through a glass wall show the expected longitudinal streaks. These are tentatively interpreted as longitudinal vortices caused by an instability of Taylor-Gortler type in the sublayer.

Exact solution of a restricted Euler equation for the velocity gradient tensor

The velocity gradient tensor satisfies a nonlinear evolution equation of the form (dAij/dt)+AikAkj− (1/3)(AmnAnm)δij=Hij, where Aij=∂ui/∂xj and the tensor Hij contains terms involving the action of cross derivatives of the pressure field and viscous diffusion of the velocity gradient. The homogeneous case (Hij=0) considered previously by Vielliefosse [J. Phys. (Paris) 43, 837 (1982); Physica A 125, 150 (1984)] is revisited here and examined in the context of an exact solution. First the equations are simplified to a linear, second‐order system (d2Aij/dt2)+(2/3)Q(t)Aij=0, where Q(t) is expressed in terms of Jacobian elliptic functions. The exact solution in analytical form is then presented providing a detailed description of the relationship between initial conditions and the evolution of the velocity gradient tensor and associated strain and rotation tensors. The fact that the solution satisfies both a linear second‐order system and a nonlinear first‐order system places certain restrictions on the soluti...

Topology of fine-scale motions in turbulent channel flow

An investigation of topological features of the velocity gradient field of turbulent channel flow has been carried out using results from a direct numerical simulation for which the Reynolds number based on the channel half-width and the centreline velocity was 7860. Plots of the joint probability density functions of the invariants of the rate of strain and velocity gradient tensors indicated that away from the wall region, the fine-scale motions in the flow have many characteristics in common with a variety of other turbulent and transitional flows: the intermediate principal strain rate tended to be positive at sites of high viscous dissipation of kinetic energy, while the invariants of the velocity gradient tensor showed that a preference existed for stable focus/stretching and unstable node/saddle/saddle topologies. Visualization of regions in the flow with stable focus/stretching topologies revealed arrays of discrete downstream-leaning flow structures which originated near the wall and penetrated into the outer region of the flow. In all regions of the flow, there was a strong preference for the vorticity to be aligned with the intermediate principal strain rate direction, with the effect increasing near the walls in response to boundary conditions.

Scale-Up Tests of High Regression Rate Paraffin-Based Hybrid Rocket Fuels

Recent research at Stanford University has led to the identification of a class of paraffin-based fuels that burn at surface regression rates that are three to four times that of conventional hybrid fuels. The approach involves the use of materials that form a thin, hydrodynamically unstable liquid layer on the melting surface of the fuel. Entrainment of droplets from the liquid-gas interface substantially increases the rate of fuel mass transfer, leading to much higher surface regression rates than can be achieved with conventional polymeric fuels. Thus, high regression rate is a natural attribute of the fuel material, and the use of oxidizing additives or other regression rate enhancement schemes is not required. The high regression rate hybrid removes the need for a complex multiport grain, and most applications up to large boosters can be designed with a single port configuration. The fuel contains no toxic or hazardous components and can be shipped by commercial freight as a nonhazardous commodity. At the present time, grains up to 0.19 m [19.1 cm (7.5 in.)] in diameter and 1.14 m [114.8 cm (45.2 in.)] long are fabricated in a general-purpose laboratory at Stanford University. To further demonstrate the feasibility of this approach, a series of scale-up tests with gaseous oxygen have been carried out using a new Hybrid Combustion Facility (HCF) at NASA Ames Research Center. Data from these tests are in agreement with the small-scale, low-pressure, and low mass flux laboratory tests at Stanford University and confirm the high regression rate behavior of the fuels at chamber pressures and mass fluxes representative of commercial applications.

A study of the fine‐scale motions of incompressible time‐developing mixing layers

The geometry of dissipating motions in direct numerical simulations (DNS) of the incompressible mixing layer is examined. All nine partial derivatives of the velocity field are determined at every grid point in the flow, and various invariants and related quantities are computed from the velocity gradient tensor. Motions characterized by high rates of kinetic energy dissipation and high enstrophy density are of particular interest. Scatter plots of the invariants are mapped out and interesting and unexpected patterns are seen. Depending on initial conditions, each type of shear layer produces its own characteristic scatter plot. In order to provide more detailed information on the distribution of invariants at intermediate and large scales, scatter plots are replaced with more useful number density contour plots. These essentially represent the unnormalized joint probability density function of the two invariants being cross‐plotted. Plane mixing layers at the same Reynolds number, but with laminar and tu...

DEVELOPMENT AND TESTING OF PARAFFIN-BASED HYBRID ROCKET FUELS

The classical hybrid combustion theory is generalized to solid fuels that form a liquid layer on their burning surface. For several classes of liquefying fuels, the layer is hydrodynamically unstable leading to substantial droplet entrainment from the melt layer into the gas stream. The susceptibility of a given fuel to this shear driven instability increases with decreasing viscosity and surface tension of the melt layer. The entrainment mass transfer, which acts in addition to the conventional gasification mechanism, is not affected by the blocking phenomenon induced by blowing from the surface. For practical oxidizer flux levels encountered in hybrid rocket applications, droplet entrainment can dominate direct gasification. Such liquefying fuels can exhibit greatly increased surface regression rates compared to classical materials such as HTPB. One application of the theory is to solid cryogenic hybrids, which utilize frozen materials for the solid propellant. The theory successfully predicts why high regression rates are observed in tests of cryogenic solid pentane, solid pentane and solid oxygen. In addition, the theory explains the dependence of the burning rates of other tested cryogenic materials on the physical properties of the liquid layer. The theory also leads to the conclusion that certain non-cryogenic materials such as paraffin and PE waxes will also exhibit high regression rates. This important result is confirmed by lab scale tests performed at Stanford University on a paraffin-based fuel.

Turbulence structures of wall-bounded shear flows found using DNS data

This work extends the study of the structure of wall-bounded flows using the topological properties of eddying motions as developed by Chong et al . (1990), Soria et al . (1992, 1994), and as recently extended by Blackburn et al . (1996) and Chacin et al . (1996). In these works, regions of flow which are focal in nature are identified by being enclosed by an isosurface of a positive small value of the discriminant of the velocity gradient tensor. These regions resemble the attached vortex loops suggested first by Theodorsen (1955). Such loops are incorporated in the attached-eddy model versions of Perry & Chong (1982), Perry et al . (1986), and Perry & Marusic (1995), which are extensions of a model first formulated by Townsend (1976). The direct numerical simulation (DNS) data of wall-bounded flows studied here are from the zero-pressure-gradient flow of Spalart (1988) and the boundary layer with separation and reattachment of Na & Moin (1996). The flow structures are examined from the viewpoint of the attached eddy hypothesis.