Unsteady Numerical Solutions Research Articles

Cooling enhancement for engine parts using jet impingement

A computational study has been performed to evaluate the use of jet impingement for cooling applications in the automotive industry. The current study uses an entire internal combustion engine cylinder with its components as a computational domain. An unsteady numerical solution for the Navier-Stokes equations was carried out using Improved Delayed Detached Eddy Simulation (IDDES). The volume of fluid approach is proposed to track and locate the liquid jet surface that is in contact with the air. The conjugate heat transfer approach is used to link the heat transfer solution between the fluid and the solid. The boundary conditions that are employed in the study are provided from lab experiments and one-dimensional simulations. The cooling jet in this study targets the hottest region in the piston, i.e., the region underneath the exhaust valve. Three nozzle sizes with flows at different Reynolds numbers are chosen to examine the thermal characteristics of the cooling jet. The computational study reveals that for a specific Reynolds number, the smaller diameter nozzle provides the highest heat transfer coefficient around the impingement point. The maximum relative velocity location at the impingement point slightly leads the location of the maximum Nusselt number. The maximum temperature in the piston decreases by 7% to 11% as the nozzle diameter changes from 1.0 to 3.0 mm for a jet Reynolds number of 4,500. If a correct selection is made for the nozzle size, the cooling jet can be efficiently used to reduce the temperature and alleviate the thermal stresses in the piston in the region underneath the exhaust valve where the maximum temperature occurs.

Unsteady temperature distribution in a cylinder made of functionally graded materials under circumferentially-varying convective heat transfer boundary conditions

Abstract A novel model on 2D unsteady conductive heat transfer in an infinite hollow cylinder is proposed. The cylinder is made of functionally graded material (FGM) that has variable properties both in radial and angular directions. Volumetric heat capacity and thermal conductivity coefficient are changed according to the power function of the radius. In the presence of variable coefficients, the governing equations of unsteady heat transfer in FGMs have caused the complexity. The Laplace transform method is used to transfer the energy equation from time to frequency domain whereas the meromorphic function is used for the inverse Laplace transform to obtain the desired solutions. The closed form solutions have been well validated and the results have been presented for different values of functionally graded indices for thermal conductivity coefficients and volumetric heat capacity. Two different FGM cases with different complicated thermal boundary conditions have been investigated. The first case has a constant temperature in the inner radius and a variable heat flux along with the convection condition in the outer radius. In the second case, the inner radius has a specific harmonic temperature and the outer radius is exposed to the convective conditions. It was observed that in both cases, the temperature value in the cylinder decreases with the increase of the FG index for the conductivity coefficient. The presented analytical solution provides a good tool for validating unsteady numerical solutions presented in the field of heat transfer in FGMs.

Theoretical Modeling of Levitated Clusters of Water Droplets Stabilized by Infrared Irradiation

Abstract This paper shows how clusters of radiation-stabilized water droplets levitated in an upward flow of air and water vapor above a heated water surface can be modeled using Spalding's self-similarity theory of heat and mass transfer and Stefan flow. The model describes equilibrium droplet states, including stability conditions, as well as nonequilibrium (quasi-steady) transient evolution. Equilibrium states are shown to exist when Stefan-flow supersaturation, which has a quadratic-like variation with height above the water surface, and radiation-stabilized equilibrium supersaturation, which is nearly constant with height, are equal. The latter can be predicted by a fundamentally derived function of absorbed radiant flux (linear), droplet radius (linear if opaque), continuum thermal conductivity, and thermodynamic properties. In fact, all of the experimentally observed droplet behavior can be predicted using simple analytical results based on quasi-steady droplet energy and continuum transport. Unsteady droplet energy, Knudsen-layer transport, numerical solutions, and curve-fitting of numerical computations, as used previously in modeling this behavior, are not necessary. An interesting reversal of the usual effect of mass transfer on droplet drag in low-Re flow when levitated droplets are irradiated asymmetrically by significant infrared radiation is also postulated, which relates to the relative importance of normal (pressure) and tangential (shear stress) drag. This theory of radiation-augmented droplet evaporation, condensation, and relative motion in a moving gas has application to conditions in clouds, wherein droplets can experience either net radiative heating or cooling and fluctuating updrafts or downdrafts.

Modal Analysis of a Laminar-Flow Airfoil under Buffet Conditions at Re = 500,000

An airfoil undergoing transonic buffet exhibits a complex combination of unsteady shock-wave and boundary-layer phenomena, for which prediction models are deficient. Recent approaches applying computational fluid mechanics methods using turbulence models seem promising, but are still unable to answer some fundamental questions on the detailed buffet mechanism. The present contribution is based on direct numerical simulations of a laminar flow airfoil undergoing transonic buffet at Mach number M = 0.7 and a moderate Reynolds number Re = 500, 000. At an angle of attack α = 4∘, a significant change of the boundary layer stability depending on the aerodynamic load of the airfoil is observed. Besides Kelvin Helmholtz instabilities, a global mode, showing the coupled acoustic and flow-separation dynamics, can be identified, in agreement with literature. These modes are also present in a dynamic mode decomposition (DMD) of the unsteady direct numerical solution. Furthermore, DMD picks up the buffet mode at a Strouhal number of St = 0.12 that agrees with experiments. The reconstruction of the flow fluctuations was found to be more complete and robust with the DMD analysis, compared to the global stability analysis of the mean flow. Raising the angle of attack from α = 3∘ to α = 4∘ leads to an increase in strength of DMD modes corresponding to type C shock motion. An important observation is that, in the present example, transonic buffet is not directly coupled with the shock motion.

Unsteady numerical solutions for a plane jet issuing from a narrow slit into a submerged space

The results of numerical simulations of a jet discharging from a narrow slit at low Reynolds numbers Re are presented. A comparison with the data of laboratory experiments is performed, and it is shown that steady numerical solutions with increasing Re transform into unsteady solutions with self-excitation of sinusoidal instability. Lateral oscillations are superimposed at the inlet section, and their influence on the jet behavior is studied.

Numerical simulation of shock-disturbances interaction in high-speed compressible inviscid flow over a blunt nose using weighted essentially non-oscillatory scheme

In the present study, shock-disturbances interaction in high-speed compressible inviscid flow is simulated utilizing the weighted essentially non-oscillatory (WENO) scheme by employing the shock-capturing technique. For this aim, the two-dimensional Euler equations in strong conservative form are discretized by using the explicit third-order TVD Runge–Kutta scheme in time and the fifth-order WENO finite difference scheme in space. The main advantage of using the WENO scheme is its capability for properly solving the discontinuities in the domain without needing any artificial viscosity, limiter function or filter. Hence, this scheme is stable, and thus, it is suitable for simulating very high-speed flows with strong shocks. Three numerical tests are presented to demonstrate the remarkable capability of the WENO scheme and the corresponding numerical results are compared with the available analytical and numerical studies in the literature. The final and main test case is the steady and unsteady numerical simulation of the high-speed compressible inviscid flow past a two-dimensional circular cylinder. For this test case, firstly, the steady numerical solution at Mach=8.03 is provided. Then, the disturbances field is added to the upstream flow field and the unsteady numerical solution is performed for the fast and slow acoustic and also entropy waves. A sensitivity study is also performed to check the effects of the grid size and the type of flux splitting method on the solution. Both the steady and unsteady numerical solutions are compared with the existing analytical and numerical solutions and some conclusions are made.

Highly Loaded Low-Pressure Turbine: Design, Numerical, and Experimental Analysis

The performance and detailed flow physics of a highly loaded, transonic low-pressure turbine stage have been investigated numerically and experimentally. The mean rotor Zweifel coefficient was 1.35, with and a total pressure ratio of 1.75. The aerodynamic design was based on recent developments in boundary-layer transition modeling. Steady and unsteady numerical solutions were used to design the blade geometry as well as to predict the design and offdesign performances. Measurements were acquired in a recently developed high-speed rotating turbine facility. The nozzle-vane-only and full-stage characteristics were measured with varied mass flow, Reynolds number, and freestream turbulences. The efficiency calculated from torque at the design speed and pressure ratio of the turbine was found to be 90.6%. This compared favorably to the mean-line target value of 90.5%. This paper will describe the measurements and numerical solutions in detail for both design and offdesign conditions.

Unsteady/Steady Hydromagnetic Convective Flow between Two Vertical Walls in the Presence of Variable Thermal Conductivity

Unsteady as well as steady natural convection flow in a vertical channel in the presence of uniform magnetic field applied normal to the flow region and temperature dependent variable thermal conductivity is studied. The nonlinear partial differential equations governing the flow have been solved numerically using unconditionally stable and convergent semi-implicit finite difference scheme. For steady case, approximate solutions have been derived for velocity, temperature, skin friction, and the rate of heat transfer using perturbation series method. Results of the computations for velocity, temperature, skin friction, and the rate of heat transfer are presented graphically and discussed quantitatively for various parameters embedded in the problem. An excellent agreement was found during the numerical computations between the steady-state approximate solutions and unsteady numerical solutions at steady-state time. In addition, comparison with previously published work is performed and the results agree well.

Efficient Prediction of Broadband Noise of a Centrifugal Fan Using U-FRPM Technique

ABSTRACT: Recently, a lot of studies have been made about the methods used to generate turbulent velocity fields stochastically in order to effectively predict broadband flow noise. Among them, the FRPM (Fast Random Particle Mesh) method which generates turbulence with specific statistical properties using turbulence kinetic energy and dissipation obtained from the steady solution of the RANS (Reynolds Averaged Navier-Stokes) equations has been successfully applied. However, the FRPM method cannot be applied to the flow noise problems involving intrinsic unsteady characteristics such as centrifugal fan. In this paper, to effectively predict the broadband noise generated by centrifugal fan, U-FR PM (unsteady FRPM) method is developed by extending the FRPM method to be combined with the unsteady numerical solutions of the unsteady RANS equations to generate the turbulence considered as broadband noise sources. Firstly, an unsteady flow field is obtained from the unsteady RANS equations through CFD (Computational Fluid Dynamics). Then, noise sources are generated using the U-FRPM method combined with acoustic analogy. Finally, the linear propagation model which is realized through BEM (Boundary Element Method) is combin ed with the generated sources to predict broadband noise at the listeners' position. The proposed technique is validated to compare its prediction result with the measured data.

Forced Flow of Vapor Condensing Over a Horizontal Plate (Problem of Cess and Koh): Steady and Unsteady Solutions of the Full 2D Problem

Accurate steady and unsteady numerical solutions of the full 2D governing equations—which model the forced film condensation flow of saturated vapor over a semi-infinite horizontal plate (the problem of Cess and Koh)—are obtained over a range of flow parameters. The results presented here are used to better understand the limitations of the well-known similarity solutions given by Koh. It is found that steady/quasisteady filmwise solution exists only if the inlet speed is above a certain threshold value. Above this threshold speed, steady/quasisteady film condensation solutions exist and their film thickness variations are approximately the same as the similarity solution given by Koh. However, these steady solutions differ from the Koh solution regarding pressure variations and associated effects in the leading part of the plate. Besides results based on the solutions of the full steady governing equations, this paper also presents unsteady solutions that characterize the steady solutions’ attainability, stability (response to initial disturbances), and their response to ever-present minuscule noise on the condensing-surface. For this shear-driven flow, the paper finds that if the uniform vapor speed is above a threshold value, an unsteady solution that begins with any reasonable initial-guess is attracted in time to a steady solution. This long time limiting solution is the same—within computational errors—as the solution of the steady problem. The reported unsteady solutions that yield the steady solution in the long time limit also yield “attraction rates” for nonlinear stability analysis of the steady solutions. The attraction rates are found to diminish gradually with increasing distance from the leading edge and with decreasing inlet vapor speed. These steady solutions are generally found to be stable to initial disturbances on the interface as well as in any flow variable in the interior of the flow domain. The results for low vapor speeds below the threshold value indicate that the unsteady solutions exhibit nonexistence of any steady limit of filmwise flow in the aft portion of the solution. Even when a steady solution exists, the flow attainability is also shown to be difficult (because of waviness and other sensitivities) at large downstream distances.

Nonlinear Stability of the Classical Nusselt Problem of Film Condensation and Wave Effects

Abstract Accurate steady and unsteady numerical solutions of the full two-dimensional (2D) governing equations for the Nusselt problem (film condensation of quiescent saturated vapor on a vertical wall) are presented and related to known results. The problem, solved accurately up to film Reynolds number of 60 (Reδ⩽60), establishes various features of the well-known steady solution and reveals the interesting phenomena of stability, instability, and nonlinear wave effects. It is shown that intrinsic flow instabilities cause the wave effects to grow over the well-known experiments-based range of Reδ⩾30. The wave effects due to film flow’s sensitivity to ever-present minuscule transverse vibrations of the condensing surface are also described. The results suggest some ways of choosing wall noise—through suitable actuators—that can enhance or dampen wave fluctuations and thus increase or decrease heat transfer rates over the laminar-to-turbulent transition zone.

The evolution of a slender non-axisymmetric drop in an extensional flow

An asymptotic method for analysing slender non-axisymmetric drops, bubbles and jets in a general straining flow is developed. The method relies on the slenderness of the geometry to reduce the three-dimensional equations to a sequence of weakly coupled, quasi-two-dimensional Stokes flow problems for the cross-sectional evolution. Exact solution techniques for the flow outside a bubble in two-dimensional Stokes flow are generalized to solve for the transverse flow field, allowing large non-axisymmetric deformations to be described. A generalization to the case where the interior of the bubble contains a slightly viscous fluid is also presented.Our method is used to compute steady non-axisymmetric solution branches for inviscid bubbles and slightly viscous drops. We also present unsteady numerical solutions showing how the eccentricity of the cross-section adjusts to a non-axisymmetric external flow. Finally, we use our theory to investigate how the pinch-off of a jet of relatively inviscid fluid is affected by a two-dimensional straining cross-flow.

Rate of unsteady reconnection in an incompressible plasma

Using a two-dimensional, incompressible magnetohydrodynamic (MHD) model, we study unsteady reconnection of antiparallel magnetic fields in the symmetric case. The magnetic reconnection process is initiated by variation of a local resistivity assumed to be a given function of time and spatial coordinates. The diffusion region of reconnection is treated as a thin boundary layer where the perpendicular components of magnetic field and velocity are much smaller than the components of these quantities along the layer. The unsteady MHD equations are solved by a two-step finite difference numerical scheme with an implicit approximation of the magnetic diffusion terms in a uniform right angle grid. In the numerical study, we use the initial conditions related to a uniform current layer without any motion of the plasma. The time-dependent reconnection process is characterized by the following important parameters: A timescale of the local resistivity, a length scale of the local resistivity, a local Reynolds number determined for the maximum resistivity and the length scale of the diffusion region. The numerical results show that the Petschek type reconnection is realized for the sufficiently small length scale andlarge time scale of the local resistivity, and a moderate local Reynolds number. Using unsteady numerical MHD solution obtained for the diffusion region, we determine the reconnection rate as a function of time by matching of the outer Petschek solution and the internal diffusion region solution.

Numerical Simulation of Acoustic Waves Interacting with a Shock Wave in a Quasi-ID Convergent-Divergent Nozzle Using an Unstructured Finite Volume Algorithm

Numerical simulations of a very small amplitude acoustic wave interacting with a shock wave in a quasi-ID convergent-divergent nozzle is performed using an unstructured finite volume algorithm with piece-wise linear, least square reconstruction, Roe flux difference splitting, and second-order MacCormack time marching. First, the spatial accuracy of the algorithm is evaluated for steady flows with and without the normal shock by running the simulation with a sequence of successively finer meshes. Then the accuracy of the Roe flux difference splitting near the sonic transition point is examined for different reconstruction schemes. Finally, the unsteady numerical solutions with the acoustic perturbation are presented and compared with linear theory results.

Axisymmetric oscillations of an opposing jet from a hemispherical nose

The flowfield around a hemispherical nose with a sonic opposing jet in a freestream of Mach 2.5 is computed by a time-accurate algorithm using the axisymmetric Navier-Stokes equations. Five total pressure ratios of a jet to a freestream, 0.816 ∼ 1.633, are examined. The results obtained strongly depend on the total pressure ratio as the previous experimental results did. In some cases, the computed flowfield oscillates intensely in which the drastic change of the jet structure and the surrounding pressure takes place. The flowfield changes from the stable flow to the unstable flow for the same total pressure ratio as in the experiment The mechanism of the oscillations and the condition of transition between both types of flow are interpreted physically on the basis of unsteady numerical solutions here.

Unsteady Finite-Analytic Method for Solute Transport in Ground-Water Flow

This work illustrates the development and application of the unsteady finite-analytic (FA) numerical solution for the migration of ground-water contamination. A functional/optimal time-weighting factor, established based on the analytic solutions of the one-dimensional, linear advection-diffusion equation, is proposed for the expression of the unsteady term in the solute-transport equation. The upstream shift nature of FA coefficients and the all-positive algebraic equation automatically originate from the FA solution, providing a physically meaningful and stable numerical scheme. Another feature of the method presented is that a very large time interval can be used in the numerical simulation. Therefore, only few time steps are needed to complete the whole computation. In the examples, the performance of unsteady FA numerical solutions is demonstrated by simulating the solute transport emitting from a Gaussian-line source. It is shown that the use of the functional time-weighting factor associated with the finite-analytic method appears to be a better choice for avoiding numerical diffusions and oscillations than, through comparison with, the fully implicit (time-weighting factor = 0) and the Crank-Nicholson (time-weighting factor = 0.5) FA schemes.

Numerical schemes for unsteady fluid flow through collapsible tubes

The study of fluid flow through compliant tubes is a fluid-structure type problem, in which a dynamic equilibrium is maintained between the fluid and the tube wall. The analogy between this flow and gas dynamics initiated the use of a number of numerical methods which were originally developed to solve compressible flow in rigid ducts. In this study we investigate the solutions obtained by applying the Lax-Wendroff and MacCormack schemes to one-dimensional incompressible flow through a straight collapsible tube. The time-evolving numerical results were compared with exact steady-state solutions. For boundary conditions which were held fixed after a prescribed rise time, the unsteady numerical solution converges to the exact steady-state solution with very good accuracy. The stability and accuracy of all the methods depend on the amount of viscous pressure loss dictated by wall friction. Flows with undamped oscillations cannot, however, be solved with these techniques.

Analysis of rate of hydrogen reduction of porous wustite pellets basing on zone-reaction models.

Porous wustite pellets were reduced with hydrogen at 900°C, and reduction curves, position of reaction zone and local fractional-reduction profiles were measured.Basic equations for the grain model were re-examined and solved as they were (unsteady numerical solution), and under quasi-steady (quasi-steady numerical one) and moreover linearization approximations (quasi-steady analytical one). When rate parameter values are selected suitably in each case, measured reduction curves and reaction zone behavior are comparatively well reproduced by the calculated results. Thiele's modulus is about 10-14 under the present experimental conditions and reaction fashion differs much from the one for unreacted-core shrinking model.When the unsteady numerical solution is calculated, reducible oxygen density is divided by M to reduce the computation time; the error at M≤2000 is within a permissible range, although the solution most faithful to the basic equations is obtained at M=1. Comparison between the unsteady and the quasi-steady numerical solutions shows that the latter is an approximate solution having rather good accuracy. The quasi-steady analytical solution is better than the others from practical viewpoint, because its computation time is the shortest and degree of agreement between the measured and the calculated results is much the same among the three.

16 カルマン渦列の自然対流による発達

Investigation in the present paper has been made theoretically and experimentally on a development of the Karman vortex street due to the natural convection in an opposing flow. Unsteady numerical solutions are obtained by means of a time-dependent numerical analysis. A development of the Karman vortex street due to the natural convection is predicted based on the result of numerical analysis. To verify the validity of the prediction, flow visualizations are carried out. The following results are obtained.1. A development of the Karman vortex street is predicted by means of a numerical analysis.2. The validity of the development is verified by means of flow visualizations.3. The mechanism of the development is made clear by the simultaneous measurements of velocity and temperature.

境界要素法による地下密度流の数値解析及び遮水壁の塩水侵入抑制の効果について

It has been reported that the salt groundwater invade inland and the aquifers become contaminated with salt as a result of over discharges of fresh groundwater by the wells in coastal areas, and the drawdown of groundwater level by the shortage of precipitation. Consequently, it is significant to calculate the movement of salt groundwater.In the present paper, for the study of preventation of salt water intrusion, the author discusses the formulation for a numerical analysis of the groundwater density flow with the boundary element method. Numerical analysis was carried out for the following two cases, (1) the steady problem in -unconfined groundwater, (2) the unsteady problem in confined groundwater.The stability of unsteady numerical solution was obtained to calculate the real part of eigen values every time step. About the behavior of fresh and salt water interface, the numerical solutions show good agreement with the experimental results.Also, the author estimated the effect of the impervious wall for countermeasure against salt water intrusion.