Equilibrium characteristics of axial-groove gas-lubricated bearings.
Reynolds equation for compressible and incompressible lubrication for load coefficients and attitude angles of axial groove cylindrical bearings
GAS BEARING↗
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An improved viscid/inviscid interaction procedure for transonic flow over airfoils
A new interacting boundary layer approach for computing the viscous transonic flow over airfoils is described. The theory includes a complete treatment of viscous interaction effects induced by the wake and accounts for normal pressure gradient effects across the boundary layer near trailing edges. The method is based on systematic expansions of the full Reynolds equation of turbulent flow in the limit of Reynolds numbers, Reynolds infinity. Procedures are developed for incorporating the local trailing edge solution into the numerical solution of the coupled full potential and integral boundary layer equations. Although the theory is strictly applicable to airfoils with cusped or nearly cusped trailing edges and to turbulent boundary layers that remain fully attached to the airfoil surface, the method was successfully applied to more general airfoils and to flows with small separation zones. Comparisons of theoretical solutions with wind tunnel data indicate the present method can accurately predict the section characteristics of airfoils including the absolute levels of drag.
Melnik, R. E.↗
On the modelling of non-reactive and reactive turbulent combustor flows
A study of non-reactive and reactive axisymmetric combustor flows with and without swirl is presented. Closure of the Reynolds equations is achieved by three models: kappa-epsilon, algebraic stress and Reynolds stress closure. Performance of two locally nonequilibrium and one equilibrium algebraic stress models is analyzed assuming four pressure strain models. A comparison is also made of the performance of a high and a low Reynolds number model for combustor flow calculations using Reynolds stress closures. Effects of diffusion and pressure-strain models on these closures are also investigated. Two models for the scalar transport are presented. One employs the second-moment closure which solves the transport equations for the scalar fluxes, while the other solves the algebraic equations for the scalar fluxes. In addition, two cases of non-premixed and one case of premixed combustion are considered. Fast- and finite-rate chemistry models are applied to non-premixed combustion. Both show promise for application in gas turbine combustors. However, finite rate chemistry models need to be examined to establish a suitable coupling of the heat release effects on turbulence field and rate constants.
Nikjooy, Mohammad↗
Face-seal lubrication: 1: Proposed and published models
The numerous published theories on the mechanism of hydrodynamic lubrication of face seals were reviewed. These theories employ either an inclined-slider-bearing macrogeometry or an inclined-slider-bearing microgeometry to produce hydrodynamic pressure that separates the surfaces of the primary seal. Secondary seal friction and primary ring inertia effects are not considered. Hypothetical seal operating models were devised to include secondary seal friction and primary ring inertia effects. It was hypothesized that these effects induce relative angular misalinement of the primary seal faces and that this misalinement is, in effect, an inclined slider macrogeometry. Stable running was postulated for some of these hypothetical operating models. In others, periodic loss of hydrodynamic lubrication was postulated to be possible with certain combinations of waviness and angular misalinement. Application of restrictions that apply to seal operation led to a hydrodynamic governing equation for the new model that is a two-dimensional, time-dependent Reynolds equation with the short-bearing approximation.
Ludwig, L. P.↗
Comparison of generalized Reynolds and Navier Stokes equations for flow of a power law fluid
This paper compares a finite element solution of a modified Reynolds equation with a finite difference solution of the Navier-Stokes equation for a power law fluid. Both the finite element and finite difference formulation are reviewed. Solutions to spiral flow in parallel and conical geometries are compared. Comparison with experimental results are also given. The effects of the assumptions used in the Reynolds equation are discussed.
Mullen, R. L.↗
A Dynamic Analysis of Hydrodynamic Wave Journal Bearings
The purpose of this paper is to study the dynamic behavior of a three-wave journal bearing using a transient approach. The transient analysis permits the determination of the rotor behavior after the fractional frequency whirl appears. The journal trajectory is determined by solving a set of nonlinear equations of motion using the Runge-Katta method. The fluid film forces are computed by integrating the transient Reynolds equation at each time step location of the shaft with respect to the bearing. Because of the large values of the rotational speeds, turbulent effects were included in the computations. The influence of the temperature on the viscosity was also considered. Numerical results were compared to experimenta1 results obtained at the NASA Glenn Research Center. Comparisons of the theoretical results with experimental data were found to be in good agreement. The numerical and experimental results showed that the fluid film of a three-wave journal bearing having a diameter of 30 mm, a length of 27 mm, and a wave amplitude ratio greater than 0.15 is stable even at rotational speeds of 60,000 RPM. For lower wave amplitude ratios, the threshold speed at which the fluid film becomes unstable depends on the wave amplitude and on the supply pocket pressure. Even if the fluid film is unstable, the wave bearing maintains the whirl orbit inside the bearing clearance.
Ene, Nicoleta M.↗
Dynamic response and stability of a gas-lubricated Rayleigh-step pad
The quasi-static, pressure characteristics of a gas-lubricated thrust bearing with shrouded, Rayleigh-step pads are determined for a time-varying film thickness. The axial response of the thrust bearing to an axial forcing function or an axial rotor disturbance is investigated by treating the gas film as a spring having nonlinear restoring and damping forces. These forces are related to the film thickness by a power relation. The nonlinear equation of motion in the axial mode is solved by the Ritz-Galerkin method as well as the direct, numerical integration. Results of the nonlinear response by both methods are compared with the response based on the linearized equation. Further, the gas-film instability of an infinitely wide Rayleigh step thrust pad is determined by solving the transient Reynolds equation coupled with the equation of the motion of the pad. Results show that the Rayleigh-step geometry is very stable for bearing number A up to 50. The stability threshold is shown to exist only for ultrahigh values of Lambda equal to or greater than 100, where the stability can be achieved by making the mass heavier than the critical mass.
Cheng, C.↗
Effects of expansion ratio on the calculation of parallel-walled backward-facing step flows - Comparison of four models of turbulence
This paper presents a numerical study of the effects of expansion ratio on two-dimensional separating and reattaching flows in plane backward-facing step geometries with parallel walls. Closure of the Reynolds equations was achieved by four different turbulence models: k-epsilon, 'modified' k-epsilon, algebraic stress, and 'modified' algebraic stress models. The k-epsilon model relates the Reynolds stresses to the mean rate of strain through the definition of an isotropic turbulent viscosity. The more advanced algebraic stress model calculates the stresses from implicit algebraic relationships containing the stresses themselves, the mean rate of strain, and the turbulent kinetic energy and its dissipation rate. 'Modified' versions of the models employ a new dissipation rate equation whose production term was made more sensitive to streamwise curvature effects. A new nonequilibrium wall function treatment proposed by Chieng and Launder (1980) was also incorporated into each model.
Sindir, M. M.↗
Renormalization group analysis of the Reynolds stress transport equation
The pressure velocity correlation and return to isotropy term in the Reynolds stress transport equation are analyzed using the Yakhot-Orszag renormalization group. The perturbation series for the relevant correlations, evaluated to lowest order in the epsilon-expansion of the Yakhot-Orszag theory, are infinite series in tensor product powers of the mean velocity gradient and its transpose. Formal lowest order Pade approximations to the sums of these series produce a fast pressure strain model of the form proposed by Launder, Reece, and Rodi, and a return to isotropy model of the form proposed by Rotta. In both cases, the model constant are computed theoretically. The predicted Reynolds stress ratios in simple shear flows are evaluated and compared with experimental data. The possibility is discussed of driving higher order nonlinear models by approximating the sums more accurately.
Rubinstein, R.↗
Starvation effects on the hydrodynamic lubrication of rigid nonconformal contacts in combined rolling and normal motion
The effect of inlet starvation on the hydrodynamic lubrication of lightly loaded rigid nonconformal contacts in combined rolling and normal motion is determined through a numerical solution of the Reynolds' equation for an isoviscous, incompressible lubricant. Starvation is effected by systematically reducing the fluid inlet level. The pressures are taken to be ambient at the inlet meniscus boundary and Reynolds' boundary condition is applied for film rupture in the exit region. Results are presented for the dynamic performance of the starved contacts in combined rolling and normal motion for both normal approach and separation. During normal approach the dynamic load ratio (i.e. ratio of dynamic to steady state load capacity) increases considerably with increase in the inlet starvation. The effect of starvation on the dynamic peak pressure ratio is relatively small. Further, it has been observed that with increasing starvation, film thickness effects become significant in the dynamic behavior of the nonconformal contacts. For significantly starved contacts the dynamic load ratio increases with increase in film thickness during normal approach and a similar reduction is observed during separation. A similar effect is noted for the dynamic peak pressure ratio.
Ghosh, M. K.↗
Starvation effects on the hydrodynamic lubrication of rigid nonconformal contacts in combined rolling and normal motion
The effect of inlet starvation on the hydrodynamic lubrication of lightly loaded rigid nonconformal contacts in combined rolling and normal motion is determined through a numerical solution of the Reynolds' equation for an isoviscous, incompressible lubricant. Starvation is effected by systematically reducing the fluid inlet level. The pressures are taken to be ambient at the inlet meniscus boundary and Reynolds' boundary condition is applied for film rupture in the exit region. Results are presented for the dynamic performance of the starved contacts in combined rolling and normal motion for both normal approach and separation. During normal approach the dynamic load ratio (i.e. ratio of dynamic to steady state load capacity) increases considerably with increase in the inlet starvation. The effect of starvation on the dynamic peak pressure ratio is relatively small. Further, it has been observed that with increasing starvation, film thickness effects become significant in the dynamic behavior of the nonconformal contacts. For significantly starved contacts the dynamic load ratio increases with increase in film thickness during normal approach and a similar reduction is observed during separation. A similar effect is noted for the dynamic peak pressure ratio.
Ghosh, M. K.↗
A New Reynolds Stress Algebraic Equation Model
A general turbulent constitutive relation is directly applied to propose a new Reynolds stress algebraic equation model. In the development of this model, the constraints based on rapid distortion theory and realizability (i.e. the positivity of the normal Reynolds stresses and the Schwarz' inequality between turbulent velocity correlations) are imposed. Model coefficients are calibrated using well-studied basic flows such as homogeneous shear flow and the surface flow in the inertial sublayer. The performance of this model is then tested in complex turbulent flows including the separated flow over a backward-facing step and the flow in a confined jet. The calculation results are encouraging and point to the success of the present model in modeling turbulent flows with complex geometries.
Shih, Tsan-Hsing↗
A turbulence model for three dimensional turbulent shear flows over curved rotating bodies
There exists currently considerable interest in the development of numerical procedures for solving the equations governing three-dimensional fluid flows. In particular, improved calculations of viscous flows on rotating bodies such a turbomachinery rotors are of great importance. The present investigation is mainly concerned with the analysis of the effect of rotation on the dissipation rate equation and the Reynolds stress equations. Attention is given to governing equations and modelling, algebraic Reynolds stress equations, a procedure for implementing the turbulence closure scheme, results for a model representative of centrifugal turbomachinery, and results for axial turbomachinery. The proposed turbulence model is valid for three-dimensional, rotating, and curved flows. The model has all the essential physics of turbulence, mean flow, rotation, and the curvature.
Galmes, J. M.↗
Renormalization group methods for the Reynolds stress transport equations
The Yakhot-Orszag renormalization group is used to analyze the pressure gradient-velocity correlation and return to isotropy terms in the Reynolds stress transport equations. The perturbation series for the relevant correlations, evaluated to lowest order in the epsilon-expansion of the Yakhot-Orszag theory, are infinite series in tensor product powers of the mean velocity gradient and its transpose. Formal lowest order Pade approximations to the sums of these series produce a rapid pressure strain model of the form proposed by Launder, Reece, and Rodi, and a return to isotropy model of the form proposed by Rotta. In both cases, the model constants are computed theoretically. The predicted Reynolds stress ratios in simple shear flows are evaluated and compared with experimental data. The possibility is discussed of deriving higher order nonlinear models by approximating the sums more accurately. The Yakhot-Orszag renormalization group provides a systematic procedure for deriving turbulence models. Typical applications have included theoretical derivation of the universal constants of isotropic turbulence theory, such as the Kolmogorov constant, and derivation of two equation models, again with theoretically computed constants and low Reynolds number forms of the equations. Recent work has applied this formalism to Reynolds stress modeling, previously in the form of a nonlinear eddy viscosity representation of the Reynolds stresses, which can be used to model the simplest normal stress effects. The present work attempts to apply the Yakhot-Orszag formalism to Reynolds stress transport modeling.
Rubinstein, R.↗
Stability of a rigid rotor supported on oil-film journal bearings under dynamic load
Most published work relating to dynamically loaded journal bearings are directed to determining the minimum film thickness from the predicted journal trajectories. These do not give any information about the subsynchronous whirl stability of journal bearing systems since they do not consider the equations of motion. It is, however, necessary to know whether the bearing system operation is stable or not under such an operating condition. The stability characteristics of the system are analyzed. A linearized perturbation theory about the equilibrium point can predict the threshold of stability; however it does not indicate postwhirl orbit detail. The linearized method may indicate that a bearing is unstable for a given operating condition whereas the nonlinear analysis may indicate that it forms a stable limit cycle. For this reason, a nonlinear transient analysis of a rigid rotor supported on oil journal bearings under: (1) a unidirectional constant load, (2) a unidirectional periodic load, and (3) variable rotating load are performed. The hydrodynamic forces are calculated after solving the time-dependent Reynolds equation by a finite difference method with a successive overrelaxation scheme. Using these forces, equations of motion are solved by the fourth-order Runge-Kutta method to predict the transient behavior of the rotor. With the aid of a high-speed digital computer and graphics, the journal trajectories are obtained for several different operating conditions.
Majumdar, B. C.↗
Numerical solution of the Navier-Stokes equations for high Reynolds number incompressible turbulent flow
The full Navier-Stokes equations for incompressible turbulent flow must be solved to accurately represent all flow phenomena which occur in a high Reynolds number incompressible flow. A two layer algebraic eddy viscosity turbulence model is used to represent the Reynolds stress in the primitive variable formulation. The development of the boundary-fitted coordinate systems makes the numerical solution of these equations feasible for arbitrarily shaped bodies. The nondimensional time averaged Navier-Stokes equations, including the turbulence mode, are represented by finite difference approximations in the transformed plane. The resulting coupled system of nonlinear algebraic equations is solved using a point successive over relaxation iteration. The test case considered was a NACA 64A010 airfoil section at an angle of attack of two degrees and a Reynolds number of 2,000,000.
Thompson, D. S.↗
Effects of Dissolution Regimes on Flow Channelization and Solute Transport in 3D Fracture Networks: Insights From Graph‐Based Reactive Transport Modeling
We investigate how mineral dissolution reshapes flow pathways and solute transport in three‐dimensional discrete fracture networks using a computationally efficient graph‐based reactive transport model. The DFNs are inspired by field‐site observations of fractured carbonate and represent realistic connectivity and structural heterogeneity. Flow is simulated with the Reynolds equation, and dissolution follows first‐order kinetics with diffusive limitations captured through an effective mass‐transfer coefficient. By systematically varying two key dimensionless parameters, the effective Damköhler number (Da), governing reaction versus advection rates, and a transport parameter (Da), analogous to the Thiele modulus, distinct flow channelization regimes emerge: mildly channelized at low G, highly channelized at intermediate Da, and extreme wormhole formation at high Da and low G. Eulerian and Lagrangian analyses, including breakthrough curves, particle tortuosity, dispersivity, and flow channeling indicators quantitatively characterize the progression of dissolution‐driven network restructuring. Across all regimes, initial fracture heterogeneity persists. The results underscore how the interplay between this initial structure, advection, reaction, and diffusion critically shapes subsurface flow pathways, with implications for applications ranging from groundwater remediation to enhanced geothermal systems.
54 ENVIRONMENTAL SCIENCES↗
Half sommerfeld approximation for finite journal bearings.
Sommerfeld approximation of oil film solution of full finite journal bearings, based on reynolds equation
SOMMERFELD APPROXIMATION↗