Unsteady Three-dimensional Numerical Simulations Research Articles

Influence of inlet shear on the 3-D flow past a square cylinder at moderate Reynolds number

Abstract Unsteady three-dimensional (3-D) numerical simulations of linear shear flow past a square cylinder at moderate Reynolds number (Re=200) are performed. The shear parameter (K) considered in this study is varied as 0.0, 0.1, and 0.2. For the uniform flow (K=0.0) case, the chosen Re falls in the transition Reynolds number range. The low frequency force pulsations of square cylinder transition phenomena are observed to decrease with increasing shear parameter. The evolution of streamwise vortical structures indicates a mode A spanwise instability in the uniform flow. Unlike in uniform flow, mixed mode A and mode B spanwise instability is observed in the case of a shear flow. The autocorrelation function of the lift and the drag coefficients is improved for any particular separation distance with increasing K.

Energy release effect of mixture on single spinning detonation structure

Unsteady three-dimensional numerical simulation on a single spinning detonation in a circular tube are presented in order to understand the effects of energy release of the mixture on the detonation structure. Overall structures of the spinning detonations such as the shock structure around the spin head, the long pressure trail, and the track angle on the wall are not affected by these effects because they depend on the specific heat ratio of the products which has approximately a constant value. The calculated averaged detonation velocities on the symmetry axis during one cycle decrease inversely with an exponential curve to become the value lower than the CJ detonation velocity. Those for p 0 = 0.1 MPa and p 0 = 0.01 MPa become approximately 0.98 D CJ and 0.92 D CJ, respectively, because the energy release in the CJ state for p 0 = 0.01 MPa is 10% lower than that for p 0 = 0.1 MPa. The state of gas behind the head of spinning detonation is also evaluated by the classical oblique shock theory and equilibrium calculation by using the track angle, shock wave angle, and detonation velocity in order to compare with the present and other researcher’s numerical results. The effects of the energy release in the mixture are large on the strength of the transverse detonation.

Effect of crystal rotation on thermocapillary flow in a shallow molten silicon pool

In order to understand the effect of crystal rotation on thermocapillary flow, we conducted a series of unsteady three-dimensional numerical simulations of thermocapillary flow in a shallow molten silicon pool with Czochralski configuration. The crucible sidewall is maintained at constant temperature. Bottom and free surfaces are adiabatic. The temperature difference between the crucible and the crystal is 16 K and crystal rotation rate varies from −10 and 10 rpm. The simulation results indicate that the spoken pattern rotates slowly with the crystal rotation at low rotation rate. When rotation rate exceeds a critical value, the spoken pattern becomes the hydrothermal waves. This transition exists hysteresis.

Effect of pool rotation on flow pattern transition of silicon melt thermocapillary flow in a slowly rotating shallow annular pool

In order to understand the mechanism of the flow pattern transitions on silicon melt in Czochralski furnaces, we conducted a series of unsteady three-dimensional numerical simulations of silicon melt flow in a slowly rotating shallow annular pool in the counter-clockwise direction. The pool was heated from the outer cylinder and cooled at the inner cylinder. The temperature differences between the vertical outer and inner cylinders ranged from 5K to 28K and annular pool rotation rate from 0 and 2rpm. Bottom and top surfaces of the melt pool were adiabatic. The simulation results indicate that two flow transitions occur when increasing the radial temperature difference along the free surface. At first, the steady two-dimensional flow becomes the first hydrothermal wave and then the second hydrothermal wave with less wave number. The critical conditions for the onset of the instability and the transition zone between the first and the second hydrothermal wave are determined at various rotation rates. Characteristics of the steady and the three-dimensional flows are discussed.

NUMERICAL INVESTIGATION OF SILICON MELT FLOW IN A SHALLOW ANNULAR POOL UNDER AN AXIAL MAGNETIC FIELD

In order to understand the effect of the magnetic field on surface patterns on semi-conducting silicon melt in industrial Czochralski furnaces, we conducted a series of unsteady three-dimensional numerical simulations of silicon melt flow in a shallow annular pool under the axial magnetic field for the magnetic field strength from 0 to 0.1T. The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces are adiabatic. When the magnetic field is weak, the simulation can predict various three-dimensional oscillatory flows depending on the radial temperature difference. With the much larger magnetic field, three-dimensional flow becomes axisymmetric steady flow. Details of flow and temperature disturbances are discussed and the critical magnetic field strengths for the onset of axisymmetric steady flow are determined.

Experimental and numerical studies on the starting effect on the secondary flow in a bend

A numerical and experimental modelling study was carried out in a curved tube to analyse the behaviour of unsteady flows in a bend. Based on a test bench, with no mechanical disturbances, the flow behaviour was observed using fluorescein injection. Velocity measurements were performed using hot-film anemometry. In addition, a finite volume method was used to perform three-dimensional unsteady numerical simulations. Womersley parameter values between 8 and 21 and Dean number values between 110 and 420 were used to assess the parameters affecting the flow behaviour. Secondary motions were observed, experimentally and numerically, showing the complexity of the flow patterns. The initiation and subsequent development are explained quantitatively. Based on our analysis of the starting effect, the secondary patterns were found to be highly dependent on both the initial conditions and the flow waveforms.

Bifurcation of thermocapillary convection in a shallow annular pool of silicon melt

In order to understand the nature of surface patterns on silicon melts in industrial Czochralski furnaces, we conducted a series of unsteady three-dimensional numerical simulations of thermocapillary convections in thin silicon melt pools in an annular container. The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces are adiabatic. The results show that the flow is steady and axisymmetric at small temperature difference in the radial direction. When the temperature difference exceeds a certain threshold value, hydrothermal waves appear and bifurcation occurs. In this case, the flow is unsteady and there are two possible groups of hydrothermal waves with different number of waves, which are characterized by spoke patterns traveling in the clockwise and counter-clockwise directions. Details of the flow and temperature disturbances are discussed and number of waves and traveling velocity of the hydrothermal wave are determined.

Three-dimensional analysis of Marangoni flow and radial segregation in Ge x Si1-x melt with Czochralski configuration

In order to examine the flow field and the radial segregation of silicon (Si) in a GexSi1-x melt with an idealized Czochralski (Cz) configuration, we conducted a series of unsteady three-dimensional (3-D) numerical simulations under zero-gravity conditions. The effect of convection driven by surface tension on the free surface of the melt was included in the model, by considering thermal, as well as solutal Marangoni convection. The concentration and flow fields at several stages during crystal growth are presented for several temperature differences, driving the Marangoni convection. The simulation results indicate that the flow and concentration fields are axisymmetric for MaT < 625 and become oscillatory and 3-D for higher values. It was found that the maximum Si concentration difference at the growth interface decreases as thermal Marangoni number increases due to higher flow velocities in the vicinity of the interface. However, temporal fluctuations of Si concentration at the interface increase at higher thermal Marangoni numbers. The effects of aspect ratio (Ar) were also considered in the model. It was found that the aspect ratio of the melt in the crucible has a prominent influence on the flow pattern in the melt which, in turn, effects the Si concentration at the growth interface.

Three-dimensional thermocapillary–buoyancy flow of silicone oil in a differentially heated annular pool

In order to understand the characteristics of thermocapillary–buoyancy flow, we conducted a series of unsteady three-dimensional numerical simulations of thermocapillary–buoyancy flow of 0.65cSt silicone oil (Prandtl number Pr = 6.7) in an annular pool with different depth ( d = 1–11 mm) heated from the outer wall (radius r o = 40 mm) and cooled at the inner cylinder ( r i = 20 mm) with an adiabatic solid bottom and adiabatic free surface. Simulation conditions correspond to those in the experiments of Schwabe [D. Schwabe, Buoyant–thermocapillary and pure thermocapillary convective instabilities in Czochralski systems, J. Crystal Growth 237–239 (2002) 1849–1853]. Simulation results with large Marangoni number predict three types three-dimensional flow patterns. In the shallow thin pool ( d = 1 mm), the hydrothermal wave characterized by curved spokes is dominant. In the deep pools ( d ⩾ 5 mm) the three-dimensional stationary flow appears and this flow pattern corresponds to the Rayleigh-Benard instability, which consists of pairs of counter-rotating longitudinal rolls. When 2 mm ⩽ d ⩽ 4 mm, the hydrothermal wave and three-dimensional oscillatory flow coexist in the pool and travel along the same azimuthal direction with the same angular velocity. The critical conditions for the onset of three-dimensional flows were determined and compared with the experimental results. The characteristics of three-dimensional flows were discussed.

Three-dimensional modelling of melt flow and segregation during Czochralski growth of Ge xSi 1− x single crystals

A series of unsteady three-dimensional (3-D) numerical simulations were carried out to examine the flow field and the radial segregation of silicon (Si) in a Ge x Si 1− x melt during Czochralski (Cz) process. In addition to the gravity driven natural convection (buoyant convection), the effect of convection driven by surface tension on the free surface of the melt was included in the model, by considering thermal, as well as solutal Marangoni convection. The concentration and flow fields during crystal growth are presented for several temperature differences ( Δ T ) , driving buoyant and Marangoni convection. It was found that the maximum silicon concentration difference at the growth interface decreases as temperature difference increases due to higher flow velocities in the vicinity of the interface. However, temporal fluctuations of Si concentration at the interface increase at higher temperature differences. The effects of aspect ratio ( A r ) were also considered in the model. It was found that the radial segregation of Si at the crystal-melt interface improves as the aspect ratio of the melt in the crucible decreases.

Three-dimensional oscillatory thermocapillary flow in encapsulated liquid bridge

In order to understand the fundamental characteristics of thermocapillary convection in an encapsulated liquid bridge, we conducted a series of unsteady three-dimensional numerical simulations of thermocapillary convection in an encapsulated liquid bridge with either 2 or 3mm inner diameter and 4–7mm outer diameter as well as variable height 1–3 mm heated from the top wall and cooled at the bottom wall with an adiabatic free surface. Simulation conditions correspond to those in the experiments of Majima [S. Majima, Phys. Fluids 13, 1517 (2001)]. The simulation results indicate that thermocapillary convection in an encapsulated liquid bridge is axisymmetric and steady at the small Marangoni number. However, when the Marangoni number exceeds some critical value, the flow will undergo a transition to three-dimensional oscillatory flow, which is characterized by a standing wave type oscillation, i.e., the periodic growth and decay of temperature disturbances from cold disk to heat disk. The critical conditions for the onset oscillatory flow are determined and compared with the experimental results. The details of the flow and temperature fields are discussed, and oscillation frequencies are also exhibited.

Three-dimensional unsteady investigation of HP turbine stages

This article deals with a three-dimensional unsteady numerical simulation of the unsteady rotor—stator interaction in a HP turbine stage. The numerical approach consists of a computational fluid dynamics (CFD) parallel code, based on an upwind total variation diminishing finite volume approach. The computation has been carried out using a sliding plane approach with hybrid unstructured meshes and a two-equation turbulent closure. The turbine rig under investigation is representative of the first stage of aeronautic gas turbine engines. A brief description of the cascade, the experimental setup, and the measuring technique is provided. Time accurate CFD computations of pressure fluctuations and Nusselt number are discussed against the experimental data.

Experimental and numerical studies on physiological flow behaviour in an asymmetric model of abdominal aortic aneurysm

An abdominal aortic aneurysm (AAA) is a localized dilatation of the abdominalaorta downstream from the renal arteries including sometimes the lilac bifurcation.Rupture is the main complication of AAAs, in 80% of cases, it leads todeath. The knowledge of the local haemodynamic, including velocity, vorticity,shear and pressure distributions, may be used to improve medical diagnosison AAA rupture.Numerical and experimental studies are therefore carried out in a three dimensionalasymmetric model of AAA with or without lilac bifurcation, to analyse thebehaviour of physiological flows in AAA. Velocity measurements are performedusing particle image velocimetry (PIV). In addition, a finite volume method [1]is used to perform three-dimensional unsteady numerical simulations. DifferentWomersley parameter values and Reynolds number values are used to assessthe parameters affecting the flow behaviour. These parameters model a normalphysiological flow rate, a moderate exercise flow rate and an intensive exerciseflow rate.For the first time, rigid walls versus compliant ones and lilac bifurcation modeldownstream from AAA versus straight artery model have been experimentallyinvestigated to analyse both the compliance and the bifurcation influences onAAA flow behaviour. Compliant wall mechanical behaviour is characterizedusing a classical traction bench [2].The secondary flow patterns and more particularly the vortices trajectoriesand their impact on the distal AAA wall are found to be highly dependent onthe flow waveforms, the wall behaviour and on the lilac bifurcation presence.These results can help to improve medical diagnosis on AAA rupture. In afuture work, we would like to numerically model the thrombus formation tobetter represent the pathological condition.

Numerical Simulation of Vortex Cavitation in a Three-Dimensional Submerged Transitional Jet

Vortex cavitation in a submerged transitional jet is studied with unsteady three-dimensional direct numerical simulations. A locally homogeneous cavitation model that accounts for non-linear bubble dynamics and bubble/bubble interactions within spherical bubble clusters is employed. The velocity, vorticity, and pressure fields are compared for both cavitating and noncavitating jets. It is found that cavitation occurs in the cores of the primary vortical structures, distorting and breaking up the vortex ring into several sections. The velocity and transverse vorticity in the cavitating regions are intensified due to vapor formation, while the streamwise vorticity is weakened. An analysis of the vorticity transport equation reveals the influence of cavitation on the relative importance of the vortex stretching, baroclinic torque, and dilatation terms. Statistical analysis shows that cavitation suppresses jet growth and decreases velocity fluctuations within the vaporous regions of the jet.

Three-dimensional thermocapillary-buoyancy flow in a shallow molten silicon pool with Cz configuration

In order to understand the characteristics of surface patterns on silicon melt in Czochralski furnaces, we conducted a series of unsteady three-dimensional numerical simulations of thermocapillary-buoyancy flow of a shallow molten silicon pool with Czochralski configuration (depth d = 3 mm). The crucible sidewall is maintained at constant temperature. Bottom and free surfaces are adiabatic or allow heat transfer in the vertical direction. The simulation results indicate that two flow transitions occur with increasing the radial temperature difference along the free surface. At first, the steady two-dimensional flow becomes steady three-dimensional flow and then oscillatory three-dimensional flow. The critical conditions for the onset of the instability were determined. Characteristics of the steady and the oscillatory three-dimensional flows were discussed.

Thermocapillary‐buoyancy flow of silicon melt in a shallow annular pool

AbstractIn order to understand the nature of surface spoke patterns on silicon melt in industrial Czochralski furnaces, a series of unsteady three‐dimensional numerical simulations were conducted for thermocapillary‐buoyancy flow of silicon melt in annular pool (inner radius ri = 15 mm, outer radius ro = 50 mm, depth d = 3 mm). The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces either are adiabatic or allow heat transfer in the vertical direction. Results show that a small temperature difference in the radial direction generates steady roll‐cell thermocapillary‐buoyancy flow. With large temperature difference, the simulation can predict three‐dimensional oscillatory flow, which is characterized by spoke patterns traveling in the azimuthal direction. The small vertical heat flux (3 W/cm2) does not have significant effects on the characteristics of this oscillatory flow. Details of the flow and temperature disturbances are discussed and the critical conditions for the onset of the oscillatory flow are determined. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Hysteretic characteristics of the Wells turbine in a deep stall condition

In order to make clear the hysteresis behaviour of Wells turbine performance in a deep stall condition, experimental and numerical investigations were made for the hysteretic characteristics of such a turbine in such conditions. The experimental investigation was made by the use of turbine equipment in which a sinusoidal flow condition was simulated. The numerical investigation was made using an unsteady three-dimensional Navier-Stokes numerical simulation. The experimental and calculated turbine unsteady performances show two characteristic hysteretic loops. The counter-clockwise hysteresis loop in the unstalled condition can be associated with the separation on the suction surface near the hub, whereas the clockwise hysteresis loop in the deep stall condition can be associated with the separation on the suction surface near the tip.

Thermocapillary flow in a shallow molten silicon pool with Czochralski configuration

In order to understand the characteristics of surface patterns on silicon melt in Czochralski (Cz) furnaces, we conducted a series of unsteady three-dimensional (3-D) numerical simulations of thermocapillary flow of a shallow molten silicon pool with Cz configuration under microgravity. The crucible sidewall is maintained at constant temperature. Bottom and free surfaces are adiabatic or allow heat transfer in the vertical direction. The simulation results indicate that two flow transitions occur when increasing the radial temperature difference along the free surface for 3-mm-deep layer. At first, the steady two-dimensional (2-D) flow becomes steady 3-D flow and then oscillatory 3-D flow. But the steady 2-D flow exhibits direct transition to an oscillatory 3-D flow for 8-mm-deep layer. The critical conditions for the onset of the instability were determined. Characteristics of the steady and the oscillatory 3-D flows were discussed.

A Stationary Vortex Phenomenon above a Low-Aspect-Ratio Wing

A stationary vortex phenomenon above a nondelta low-aspect-ratio wingwas obtained in three-dimensional unsteady numerical simulation. Flowvisualization is conducted in water channel using hydrogen bubbles. Theresults verify that there is a vortex trapped above thelow-aspect-ratio wing and the stationary vortex consisted of twosemi-balls and anti-rotation vortices which are different from theleading edge vortices on the delta wing.

Three-dimensional numerical simulation of thermocapillary flow of moderate Prandtl number fluid in an annular pool

A series of unsteady three-dimensional numerical simulations were conducted for thermocapillary flow of 0.65 cSt silicone oil (Prandtl number Pr=6.7) in annular pools with different depths ( d=1–17 mm) heated from the outer wall (radius r o=40 mm) and cooled at the inner cylinder ( r i=20 mm) with an adiabatic solid bottom and adiabatic free surface. Simulation conditions corresponded to those in the microgravity experiments of Schwabe and Benz (Adv. Space Res. 29 (2002) 629). A temperature difference between the outer and inner walls (Δ T) generates roll-cell thermocapillary flow. Simulations with large Δ T predicted two types of oscillatory thermocapillary flows. One is the hydrothermal wave with curved spokes, which is dominant only in shallow pools ( d=1–3 mm). The second one is dominant in deep pools ( d>3 mm) and is characterized by traveling straight spoke patterns in the azimuthal direction. Critical conditions for the onset of the oscillatory flow in liquid pools of different depths were determined. Details of the flow and temperature fields were discussed. Oscillation frequencies were compared with the experimental results as functions of Δ T and pool depth d. Quantitative agreements were obtained for small Δ T and shallow pools.