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.

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 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.

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.

Unsteady Flow Simulations in a Three-lobe Positive Displacement Blower

To improve the performance of the positive displacement blower, it is imperative to understand the detailed internal flow characteristics or enable a visualization of flow status. However, the existing two-dimensional unsteady, three-dimensional steady or quasi-unsteady numerical simulation and theoretical analysis cannot provide the detailed flow information, which is unfavorable to improve the performance of positive displacement blower. Therefore, the unsteady flow characteristics in a three-lobe positive displacement blower are numerically investigated by solving the three-dimensional, unsteady, compressible Navier-Stokes equations coupled with RNG k-e turbulent model. In the numerical simulation, the dynamic mesh technique and overset mesh updating method are adopted. Due to the air being compressed in the process of the rotors rotating, the variation of the temperature field in the positive displacement blower is considered. By comparing the experimental measurements and the numerical results on the variation of flow rate with the outlet pressure, the maximum relative error of the flow rate is less than 2.15% even at the maximum outlet pressure condition, which means that the calculation model and numerical computational method used are effective. The numerical results show that in the intake region, the fluctuations of the inlet flow are greatly affected by the direction of the velocity vectors. In the exhaust region, the temperature changes significantly, which leads to the increase of the airflow pulsation. Through analysis on the velocity, pressure and temperature fields obtained from the numerical simulations, three-dimensional unsteady flow characteristics in the positive displacement blower are revealed. The studied results will provide useful reference for improving the performance and empirical correction in the design of the positive displacement blower.

The Vibration Controllability of 20~# Steel Pipe Excited by Unsteady Flow

An excited experiment system of 20# steel pipe was established with oil cylinder, 20# steel pipe, frequency converter, pump station and wave exciter generating unsteady flow artificially. The experimental results showed that the 20# steel pipe could vibrate with the excitation of unsteady flows, and the vibration was periodic, instead of a harmonic one. Particles on the front and rear positions of pipe vibrated synchronously, and the vibration intensity of the pipe＇s two ends was greater than in the middle. System pressure and wave exciter＇s frequency had much influence upon pipe＇s amplitude. Pipe＇s vibration frequency was little affected by system pressure, and its value was close to the wave exciter＇s. Therefore, the active control of pipe＇s vibration can be realized by setting system pressure and adjusting frequency converter＇s frequency.

Computational fluid dynamics of left ventricular assist device under unsteady flow

Left ventricular assist device( LVAD) in this study is a mechanical tool that is used to support blood flow in the patient with heart disease. It supports left ventricle by building up the pressure to the pump outlet connected to the aorta. This pump was designed based on the magnetic driven centrifugal pump with a unique small washout hole constructed inside the impeller to generate the washout flow passage to prevent the stagnation at the region underneath and around the rotor. Computational fluid dynamics( CFD) was adopted in this study to assess the performance and optimize the design to avoid recirculation and high shear stress which is the main cause of stagnation and blood damage. Transient simulation was used for this study due to the asymmetric design of the washout hole and the complication of the bottom support of the impeller that has a risk of thrombosis,also,it was used to predict the variation of hydraulic performance caused by the rotation of the impeller and pulsed flow at the pump inlet. The simulation results show no excessive stress and no recirculation observed within the computational domain; in addition,the research result also provides information for further optimization and development to the pump.

LES Analysis of the Unsteady Flow Characteristics of a Centrifugal Pump Impeller

Stall phenomena increase the complexity of the internal flow in centrifugal pump impellers.In order to tackle this problem,in the present work,a large eddy simulation(LES)approach is applied to determine the characteristics of these unstable flows.Moreover,a vorticity identification method is used to characterize quantitatively the vortex position inside the impeller and its influencing area.By comparing the outcomes of the numerical simulations and experimental results provided by a Particle Image Velocimetry(PIV)technique,it is shown that an apparent“alternating stall”phenomenon exists inside the impeller when relatively small flow rate conditions are considered.The stall is generated near the suction side of the blade inlet,grows towards the high-pressure side of the blade in the circumferential direction,and gradually attenuates.As the flow rate decreases,the number of stalls remains unchanged,while the related influencing area and strength gradually increase and the circumferential velocity increases.

THEORETICAL DISCUSSION ON UNSTEADY FLOW OF FLUID DURING METAL ATOMIZATION

The expressions of fluid velocity under several conditions during gas atomization according to the unsteady flow mechanics are propsed, and the basic ways to maintain the stability of fluid flow is analyzed

A numerical simulation of unsteady flow in small diameter helical grafts

The application of small diameter arterial grafts is limited due to the fact of relatively poor long-time patency which is caused by thrombosis formation in the short term and intimal hyperplasia(IH) in the medium and long term.Thrombosis,obstructing the flow of blood

ANALYTICAL SOLUTIONS FOR EQUATIONS OF UNSTEADY FLOW OF NON-NEWTONIAN FLUIDS IN TUBE

This paper presents analytieal solutions to the partial differential equations for unsteady flow of the second-order fluid and Maxwell fluid in tube by using the integral transform method. It can be used to analyse the behaviour of axial velocity and shear stress for unsteady flow of nun-Newtonian visco-elastie fluids in tube, and to provide a theoretical base for the projection of pipe-line engineering.

UNSTEADY FLOW ANALYSIS IN A CROSS-FLOW FAN WITH A CASING

A numerical analysis method has been developed for the prediction of the flow in across-flow fan with a casing. The method has no simplifying assumptions regarding the geometryof both the impeller blades and the casing Taking into consideration the effect of the boundarylayer on the upper wall surface of the casing, the internal flow is analyzed using both the directboundary element method and the discrete vortex method by assuming that the flow istwo-dimensional. inviscid and incomtressible. In order to satisfy Kelven's theorem, vorticity is sh-ed at the trailing edges of each impeller blade, then the discrete vortex method is used to approxi-mate the convection of the free vorticity. Results are giver for the cross-flow fan with two differ-ent configurations of the casing with an dentical impeller.

DISCONTINUITY-CAPTURING FINITE ELEMENT COMPUTATION OF UNSTEADY FLOW WITH ADAPTIVE UNSTRUCTURED MESH

A discontinuity-capturing scheme of finite element method(FEM)is proposed.The unstructured-grid technique combined with a new type of adaptive mesh approach is developed for both compressible and incompressible unsteady flows,which exhibits the capability of capturing the shock waves and/or thin shear layers accurately in an unsteady viscous flow at high Reynolds number. In particular,a new testing variable,i.e.,the disturbed kinetic energy E,is suggested and used in the adaptive mesh computation,which is universally applicable to the capturing of both shock waves and shear layers in the inviscid flow and viscous flow at high Reynolds number.Based on several calculated examples,this approach has been proved to be effective and efficient for the calculations of compressible and incompressible flows.

A Confined Laminar Slot Impinging Jet at Low Reynolds Numbers:Unsteady Flow and Heat Transfer Characteristics

In this study,the unsteady flow and heat transfer characteristics of a laminar slot jet at low Reynolds numbers impinging on an isothermal plate surface in a two-dimensional confined space are numerically investigated.The investigations are performed at Reynolds numbers of 120,150 and 200 based on the nozzle width and mean inlet velocity of the jet.Results show that the Reynolds numbers of 120,150 and 200 correspond to different flow features,namely,a steady flow,an intermittent flapping motion of jet column and a continuous sinusoidal flapping state,respectively.Based on some time snapshots of the flow field,the dynamic characteristics and driving mechanism of the intermittent flapping motion of the jet column and the continuous sinusoidal flapping state are explained.When the jet flaps at the Reynolds number 150 and 200,there are other Nusselt number peaks outside the stagnation zone,which are related to the interference between the vortices shedding on both sides of the jet and the boundary layers of the plate surface.Furthermore,the dynamic mode decomposition is implemented to accurately extract flow modes with characteristic frequencies.For a Reynolds number of 150,there is a flapping mode,which describes the lateral flapping motion of the jet column.When the Reynolds number is 200,there are multiple modes related to the flapping motion of the jet,as well as a low-frequency mode,which reflects the periodic changes of the boundary contour and position of the recirculation zone.

UNSTEADY FLOW ANALYSIS AND EXPERIMENTAL INVESTIGATION OF AXIAL-FLOW PUMP

The three-dimensional unsteady turbulent flow in axial-flow pumps was simulated based on Navier-Stoke solver embedded with k - ε RNG turbulence model and SIMPLEC algorithm. Numerical results show that the unsteady prediction results are more accurate than the steady results, and the maximal error of unsteady prediction is only 4.54%. The time-domain spectrums show that the static pressure fluctuation curves at the inlet and outlet of the rotor and the outlet of the stator are periodic, and all have four peaks and four valleys. The pressure fluctuation amplitude increases from the hub to the tip at the inlet and outlet of the rotor, but decreases at the outlet of the stator. The pressure fluctuation amplitude is the greatest at the inlet of the rotor, and the average amplitude decreases sharply from the inlet to the outlet. The frequency spectrums obtained by Fast Fourier Transform （FFT） show that the dominant frequency is approximately equal to the blade passing frequency. The static pressure on the pressure side of hydrofoil on different stream surfaces remains almost consistent, and increases gradually from the blade inlet to the exit on the suction side at different time steps. The axial velocity distribution is periodic and is affected by the stator blade number at the rotor exit. The experimental results show that the flow is almost axial and the pre-rotation is very small at the rotor inlet under the conditions of 0.8 QN -1.2 QN Due to the clearance leakage, the pressure, circulation and meridional velocity at the rotor outlet all decrease near the hub leakage and tip clearance regions.

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.

STUDY AND APPLICATION OF STEADY FLOW AND UNSTEADY FLOW MATHEMATICAL MODEL FOR CHANNEL NETWORKS

Based on the Preissmann implicit scheme for the one-dimensional Saint-Venant equation, the mathematical model for one-dimensional fiver networks and canal networks was developed and the key issues on the model were expatiated particularly in this article. This model applies the method of three-steps solution for chaunel-junction-channel to simulate the river networks, and the Gauss elimination method was used to calculate the sparse matrix. This model was applied to simulate the tree-type irrigation canal networks, complex looped channel networks and the Lower Columbia Slough networks. The results of water level and discharge agree with the data from the Adlul and field data. The model is proved to be robust for simulating unsteady flows in river networks with various degrees of complex structure. The calculated results show that this model is useful for engineering applications in complicated river networks. Future research was recommended to focus on setting up ecological numerical model of water quality in river networks and canal networks.

REAL-TIME FLOOD FORECASTING METHOD WITH 1-D UNSTEADY FLOW MODEL

A real-time forecasting method coupled with the I-D unsteady flow model with the recursive least-square method was developed. The 1-D unsteady flow model was modified by using the time-variant parameter and revising it dynamically through introducing a variable weighted forgetting factor, such that the output of the model could be adjusted for the real time forecasting of floods. The application of the new real time forecasting model in the reach from Yichang to Luoshan of the Yangtze River was demonstrated. Computational result shows that the forecasting accuracy of the new model is much higher than that of the original 1-D unsteady flow model. The method developed is effective for flood forecasting, and can be used for practical operation in the flood forecasting.

PRESSURE DISTRIBUTION ON THE WALL OF THE INNER CYLINDER RECIPROCATING AXIALLY TO THE UNSTEADY FLOW OF VISCOELASTIC FLUID IN ECCENTRIC ANNULUS

In this article, the governing equations for the unsteady flow of viscoelastic fluid in the eccentric annulus with the inner cylinder reciprocating axially and the expression of the pressure distribution on the wall of the inner cylinder of the annulus are established and derived, respectively, under the bipolar coordinate system. The equations and the expression are solved and calculated numerically using the finite difference method, respectively. The curves of the pressure distribution on the wall of the inner cylinder of the aqueous solution of Hydrolyzed Polyacrylamide （HPAM） are plotted and the influences of annular eccentricity, stroke, and stroke frequency on the pressure distribution are analyzed.