On the analytical solution of transient friction in channel flows

The transient friction in channel mean flows is the sum of two contributions,i.e.,the underlying laminar flow(ULF)and the purely turbulent component(PTC),and the contributions are analyzed separately by theoretical experiments.It is found that,the transient friction may be higher or remarkably lower than that in equal-Reynolds number steady-state flows.The universal time constant for plane-parallel laminar flows is reported,and the role of the time constant in a turbulent mean flow is examined.It is shown that the time constant is related to the turbulence's frozen time.Finally,a study of the logarithmic layer during the transient flow is accomplished,which shows that the logarithmic layer is destroyed.

Finite Difference Approach on Magnetohydrodynamic Stratified Fluid Flow Past Vertically Accelerated Plate in Porous Media with Viscous Dissipation

This study intends to evaluate the influence of temperature stratification on an unsteady fluid flow past an accelerated vertical plate in the existence of viscous dissipation.It is assumed that the medium under study is a grey,non-scattered fluid that both fascinates and transmits radiation.The leading equations are discretized using the finite differencemethod(FDM).UsingMATLABsoftware,the impacts of flowfactors on flowfields are revealed with particular examples in graphs and a table.In this regard,FDM results show that the velocity and temperature gradients increase with an increase of Eckert number.Furthermore,tables of the data indicate the influence of flow-contributing factors on the skin friction coefficients,and Nusselt numbers.When comparing constant and variable flow regimes,the constant flow regime has greater values for the nondimensional skin friction coefficient.This research is both innovative and fascinating since it has the potential to expand our understanding of fluid dynamics and to improve many different sectors.

Numerical Investigation of the Unsteady Flow in a Transonic Compressor with Curved Rotors

The unsteady 3D flow fields in a single-stage transonic compressor under designed conditions are simulated numerically to investigate the effects of the curved rotors on the stage performance and the aerodynamic interaction between the blade rows. The results show that, compared to the compressor with unurved rotors, the compressor under scrutiny acquires remarkable increases in efficiency with significantly reduced amplitudes of the time-dependent fluctuation. The amplitude of the pressure fluctuation around the stator leading edge decreases at both endwalls, but increases at the mid-span in the curved rotors. The pressure fluctuation near the stator leading edge, therefore, becomes more uniform in the radial direction of this compressor. Except for the leading edge area, the pressure fluctuatinn amplitude declines remarkably in the tip region of stator surface downstream of the curved rotor, but hardly changes in the middle and at the hub.

GRIDLESS METHOD FOR UNSTEADY VISCOUS FLOWS

Gridless method is developed for unsteady viscous flows involving moving boundaries. The point distri- bution of gridless method is implemented in an isotropic or anisotropic way according to the features of viscous flows. In the area far away from the body, the traditional cloud of isotropic points is used, while in the adjacent area, the cloud of anisotropic points is distributed. In this way, the point spacing normal to the wall can be small enough for simulating the boundary layer, and meanwhile, the total number of points in the computational do- main can be controlled due to large spacing in other tangential direction through the anisotropic way. A fast mov- ing technique of clouds of points at each time-step is presented based on the attenuation law of disturbed motion for unsteady flows involving moving boundaries. In the mentioned cloud of points, a uniform weighted least- square curve fit method is utilized to discretize the spatial derivatives of the Navier-Stokes equations. The pro- posed gridless method, coupled with a dual time-stepping method and the Spalart-Allmaras turbulence model, is implemented for the Navier-Stokes equations. The computational results of unsteady viscous flows around a NLR7301 airfoil with an oscillating flap and a pitching NACA0012 airfoil are presented in a good agreement with the available experimental data.

UNSTEADY FLOWS OF A GENERALIZED SECOND GRADE FLUID WITH THE FRACTIONAL DERIVATIVE MODEL BETWEEN TWO PARALLEL PLATES

The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced.Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus.The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates.In addition,the solutions of the shear stresses at the plates are also determined.

Numerical simulation and analysis of solid-liquid two-phase threedimensional unsteady flow in centrifugal slurry pump

Based on RNG k-ε turbulence model and sliding grid technique, solid-liquid two-phase three-dimensional(3-D) unsteady turbulence of full passage in slurry pump was simulated by means of Fluent software. The effects of unsteady flow characteristics on solid-liquid two-phase flow and pump performance were researched under design condition. The results show that clocking effect has a significant influence on the flow in pump, and the fluctuation of flow velocity and pressure is obvious, particularly near the volute tongue, at the position of small sections of volute and within diffuser. Clocking effect has a more influence on liquid-phase than on solid-phase, and the wake-jet structure of relative velocity of solid-phase is less obvious than liquid-phase near the volute tongue and the impeller passage outlet. The fluctuation of relative velocity of solid-phase flow is 7.6% smaller than liquid-phase flow at the impeller outlet on circular path. Head and radial forces of the impeller are 8.1% and 85.7% of fluctuation, respectively. The results provide a theoretical basis for further research for turbulence, improving efficient, reducing the hydraulic losses and wear. Finally, field tests were carried out to verify the operation and wear of slurry pump.

Studies of the unsteady supersonic base flows around three afterbodies

Unsteady supersonic base flows around three afterbodies, cylindrical （Cy）, boattailed （BT） and three-step （MS）, are investigated in this paper. Reynolds-averaged Navier-Stokes （RANS） and two RANS/LES （large-eddy simulation） hybrid methods, detached eddy simulation （DES） and delayed-DES （DDES）, are used to predict the base flow characteristics around the baseline Cy afterbody. All the RANS and hybrid methods are based on the two-equation SST （shear-stress transport） model with compressible corrections （CC）. According to the comparison of measurements, both DES and DDES can produce more satisfactory results than RANS. RANS can only present the ＂stable＂ flow pat- terns, while the hybrid methods can demonstrate unsteady flow structures. DDES and DES results are little different from one another although the latter exhibits better agreement with the experiment. DES is taken to investigate the 5° BT and three-step afterbodies. The mean flow data and the instantaneous turbulent coherent structures are compared against available measurements.

Generation of Dynamic Grids and Computation of Unsteady Transonic Flows around Assemblies

Algebraic methods and rapid deforming techniques are used to generate three-dimensional boundary-fitted dynamic grids for assemblies. The conservative full-potential equation is solved by a time-accurate approximate factorization algorithm and internal Newton iterations. An integral boundary layer method based on the dissipation integral is used to account for viscous effects. The computational results about unsteady transonic forces on wings, bodies and control surfaces are in agreement with experimental data.

Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders

The fractional calculus is used in the constitutive relationship model of viscoelastic fluid. A generalized Maxwell model with fractional calculus is considered. Based on the flow conditions described, two flow cases are solved and the exact solutions are obtained by using the Weber transform and the Laplace transform for fractional calculus.

Gridless Solution Method for Two-Dimensional Unsteady Flow

The main purpose of this paper is to develop a gridless method for unsteady flow simulation. A quadrantal point infilling strategy is developed to generate point and combine clouds of points automatically. A point-moving algorithm is introduced to ensure the clouds of points following the movements of bodyboundaries. A dual time method for solving the two-dimenslonal Euler equations in Arbitrary Lagrangian-Eulerian （ALE） formulation is presented. Dual time method allows the real-time step to be chosen on the basis of accuracy rather than stability. It also permits the acceleration techniques, which are commonly used to speed up steady flow calculations, to be used when marching the equations in pseudo time. The spatial derivatives, which are used to estimating the inviscid flux, are directly approximated by using local least-squares curve method. An explicit multistage Runge-Kutta algorithm is used to advance the flow equations in pseudo time. In order to accelerate the solution to convergence, local time stepping technique and residual averaging are employed. The results of NACA0012 airfoil in transonic steady flow are presented to verify the accuracy of the present spatial discretization method. Finally, two AGARD standard test cases in which NACA0012 airfoil and NACA64A010 airfoil oscillate in transonic flow are simulated. The computational results are compared with the experimental data to demonstrate the validity and practicality of the presented method.

An SOR Implicit Time-accurate Scheme in Calculating Unsteady Flows

A new time-accurate marching scheme for unsteady flow calculations is proposed in the present work. This method is the combination of classical Successive Over-Relaxation （SOR） iteration method and Jacobian matrix diagonally dominant splitting method of LUSGS. One advantage of this algorithm is the second-order accuracy because of no factorization error. Another advantage is the low computational cost because the Jacobian matrices and fluxes are only calculated once in each physical time step. And, the SOR algorithm has better convergence property than Gauss-Seidel. To investigate its accuracy and convergency, several unsteady flow computational tests are carded out by using the proposed SOR algorithm. Roe＇s FDS scheme is used to discritize the inviscid flux terms. Unsteady computational results of SOR are compared with the experiment results and those of Gauss-Seidel, Results reveal that the numerical results agree well with the experimental data and the second-order accuracy can be obtained as the Gauss-Seidel for unsteady flow computations. The impact of SOR factor is investigated for unsteady computations by using different SOR factors in this algorithm to simulate each computational test. Different numbers of inner iterations are needed to converge to the same criterion for different SOR factors and optimal choice of SOR factor can improve the computational efficiency greatly.

A Lagrangian criterion of unsteady flow separation for two-dimensional periodic flows

The present paper proposes a Lagrangian criterion of unsteady flow separation for two-dimensional periodic flows based on the principle of weighted averaging zero skin-friction given by Haller （HALLER, G. Exact theory of unsteady separation for two-dimensional flows. Journal of Fluid Mechanics, 512, 257-311 （2004））. By analyzing the distribution of the finite-time Lyapunov exponent （FTLE） along the no-slip wall, it can be found that the periodic separation takes place at the point of the zero FTLE. This new criterion is verified with an analytical solution of the separation bubble and a numerical simulation of lid-driven cavity flows.

An application of interacting shear flows theory: exact solution for unsteady oblique stagnation point flow

An analytical solution of the governing equations of the interacting shear flows for unsteady oblique stagnation point flow is obtained. It has the same form as that of the exact solution obtained from the complete NS equations and physical analysis and relevant discussions are then presented.

A note on the Galilean invariance of aerodynamic force theories in unsteady incompressible flows

As a basic principle in classical mechanics,the Galilean invariance states that the force is the same in all inertial frames of reference.But this principle has not been properly addressed by most unsteady aerodynamic force theories,if the partial force contributed by a local flow structure is to be evaluated.In this note,we discuss the Galilean-invariance conditions of the partial force for several typical theories and numerically test what would happen if these conditions do not hold.

Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.

Computation of Unsteady Flow Past a Biomimetic Fin

The unsteady hydrodynamics of a biomimetic fin attached to a cylindrical body has been studied numerically using a computational fluid dynamic (CFD) simulator based on an in-house solver of the Navier-Stokes equations, combined with a recently developed multi-block, overset grid method. The fin-body CFD model is based on a mechanical pectoral fin device, which consists of a cylindrical body and an asymmetric fin and can mimic flapping, rowing and feathering motions of the pectoral fins in fishes. First the multi-block, overset grid method incorporated into the NS solver was verified through an extensive study of unsteady flows past a single fin undergoing rowing and feathering motion. Then unsteady flows past the biomimetic fin-body model undergoing the same motions were computed and compared with the measurements of forces of the mechanical pectoral fin, which shows good agreement in both time-varying and time-averaged hydrodynamic forces. The relationship between force generation and vortex dynamics points to the importance of the match in fin kinematics between power and recovery strokes and implies that an optimal selection of parameters of phase lags between and amplitudes of rowing and feathering motions can improve the performance of labriform propulsion in terms of either maximum force generation or minimum mechanical power.

Boundary Element Analysis (Laplace Transform Solution) of Groundwater Unsteady Flow to a Multiple Well System in a Confined Aquifer

The calculations of unsteady flow to a multiple well system with the application of boundary elementmethod (BEM) are discussed. The mathematical model of unsteady well flow is a boundary value problem ofparabolic differential equation. It is changed into an elliptic one by Laplace transform to eliminate time varia-ble. The image function of water head H can be solved by BEM. We derived the boundary integral equation ofthe transformed variable H and the discretization form of it, so that there is no need to discretize the bounda-ries of well walls and it becomes easier to solve the groundwater head H by numerical inversion.

Experimental Study on Calculation of Hydro-Geological Parameters for Unsteady Flow

After the linear analytical method of unsteady flow theory is further improved,an innovative and faster algorithm is introduced.The water storage in a confined aquifer is derived from the water transmissivity coefficient and the water-pressure conductivity coefficient.The water transmissivity coefficient is approximated by a Taylorseries expansion of drawdown,and the water-pressure conductivity coefficient is obtained by the average drawdown.In this algorithm,the distance of the observation points from the pumping well must be short.When the distance is as short as the radius of the main pumping well,the data of the drawdown difference between the sidewall and the center of pumping well are difficult to measure,but the same results can be achieved based on the assumption that the drawdown difference approximates to the drawdown of the observation wells at a radial distance from the pumping well according to the algorithm.Without the help of charts,this algorithm is more concise and efficient,which has been verified by the test of water pumping project in Tianjin Binhai International Airport.

Entropy analysis of unsteady magneto-nanofluid flow past accelerating stretching sheet with convective boundary condition

The unsteady laminar magnetohydrodynamics （MHD） boundary layer flow and heat transfer of nanofluids over an accelerating convectively heated stretching sheet are numerically studied in the presence of a transverse magnetic field with heat source/sink The unsteady governing equations are solved by a shooting method with the Runge-Kutta- Fehlberg scheme. Three different types of water based nanofluids, containing copper, aluminium oxide, and titanium dioxide, are taken into consideration. The effects of the pertinent parameters on the fluid velocity, the temperature, the entropy generation num- ber, the Bejan number, the shear stress, and the heat transfer rate at the sheet surface are graphically and quantitatively discussed in detail. A comparison of the entropy generation due to the heat transfer and the fluid friction is made with the help of the Bejan number. It is observed that the presence of the metallic nanoparticles creates more entropy in the nanofluid flow than in the regular fluid flow.