Tangential Gradient Research Articles

Self-propulsion in 2D confinement: phoretic and hydrodynamic interactions.

Chemically active Janus particles generate tangential concentration gradients along their surface for self-propulsion. Although this is well studied in unbounded domains, the analysis in biologically relevant environments such as confinement is scarce. In this work, we study the motion of a Janus sphere in weak confinement. The particle is placed at an arbitrary location, with arbitrary orientation between the two walls. Using the method of reflections, we study the effect of confining planar boundaries on the phoretic and hydrodynamic interactions, and their consequence on the Janus particle dynamics. The dynamical trajectories are analyzed using phase diagrams for different surface coverage of activity and solute-particle interactions. In addition to near wall states such as 'sliding' and 'hovering', we demonstrate that accounting for two planar boundaries reveals two new states: channel-spanning oscillations and damped oscillations around the centerline, which were characterized as 'scattering' or 'reflection' by earlier analyses on single wall interactions. Using phase-diagrams, we highlight the differences in inert-facing and active-facing Janus particles. We also compare the dynamics of Janus particles with squirmers for contrasting the chemical interactions with hydrodynamic effects. Insights from the current work suggest that biological and artificial swimmers sense their surroundings through long-ranged interactions, that can be modified by altering the surface properties.

A fully Eulerian hybrid immersed boundary-phase field model for contact line dynamics on complex geometries

We present a fully Eulerian hybrid immersed-boundary/phase-field model to simulate wetting and contact line motion over any arbitrary geometry. The solid wall is described with a volume-penalisation ghost-cell immersed boundary whereas the interface between the two fluids by a diffuse-interface method. The contact line motion on the complex wall is prescribed via slip velocity in the momentum equation and static/dynamic contact angle condition for the order parameter of the Cahn-Hilliard model. This combination requires accurate computations of the normal and tangential gradients of the scalar order parameter and of the components of the velocity. However, the present algorithm requires the computation of averaging weights and other geometrical variables as a preprocessing step. Several validation tests are reported in the manuscript, together with 2D simulations of a droplet spreading over a sinusoidal wall with different contact angles and slip length and a spherical droplet spreading over a sphere, showing that the proposed algorithm is capable to deal with the three-phase contact line motion over any complex wall. The Eulerian feature of the algorithm facilitates the implementation and provides a straight-forward and potentially highly scalable parallelisation. The employed parallelisation of the underlying Navier-Stokes solver can be efficiently used for the multiphase part as well. The procedure proposed here can be directly employed to impose any types of boundary conditions (Neumann, Dirichlet and mixed) for any field variable evolving over a complex geometry, modelled with an immersed-boundary approach (for instance, modelling deformable biological membranes, red blood cells, solidification, evaporation and boiling, to name a few).

Magnetized flow of sutterby nanofluid through cattaneo-christov theory of heat diffusion and stefan blowing condition

Stefan blowing phenomenon in electrically conducting Sutterby material flow over stretchable rotating disk is demonstrated in this research. Cattaneo-Christov (CC) model of energy diffusion is adopted to analyze the heat transmission. Buongiorno model is carried out to evaluate the involvement of nanoparticles. The formulated system of partial differential expressions is re-structured by the enactment of similarity functions. Runge–Kutta-Fehlberg (RKF) fourth-fifth order process has been executed to communicate the solution of velocity, thermal and solutal fields. The velocity, concentration, thermal fields, skin friction, rate of mass and heat transportations are explored for the embedded non-dimensional parameters graphically. Result reveals that the rise in Stefan blowing factor leads to an enhancement in radial and tangential velocities gradients. The velocity of nanomaterial is reduced by the incrementing material parameter values. The augmenting magnetic parameter values reduced the liquid velocity but improves the temperature. The thermophoretic force and Brownian motion involvement resulted the higher thermal field.

Experimental study on the influence of tangential stress gradient on the energy evolution of strainburst

Strainburst is a kind of safety concern caused by the release of elastic strain energy from surrounding rocks under a tangential stress gradient. In this paper, laboratory rockburst model tests under four different stress gradients were conducted. Based on the acoustic emission (AE) data, the influence of tangential stress gradient on the energy evolution of strainburst was studied. The results indicate that (1) by controlling the tangential stress gradient loading at the top of the specimen, the change process of tangential and radial stress gradient from surrounding rocks caused by excavation disturbance, to a large extent, can be simulated. (2) The failure phenomena and failure stress of the specimen are both related to its tangential stress gradient distribution. As the tangential stress gradient increases, the failure stress is reduced, but dynamic failure phenomena become more evident. (3) With the increase of tangential stress gradient, the accumulation period of elastic strain energy lengthens, whereas the dissipation and release periods shorten during the loading process. As the tangential stress gradient rises, when the specimen rockburst occurs, the more dramatic decrease and faster decreasing rate of AE b-value reveal the increase in the specimen’s proportion of shear failure. Also, the critical index (r) of probability density distribution shows a downward trend, which indicates that the AE energy in the low disturbance area decreases. In contrast, the AE energy span in the high disturbance area widens and the energy level becomes higher.

Hydrothermal waves in sessile droplets evaporating at a constant contact angle mode

Although various Marangoni instability patterns have been observed in evaporating sessile droplets, most are limited to a small contact angle. As a result, the droplets resemble a flat liquid layer, where the vertical temperature gradient dominates, and the hydrothermal waves (HTWs) predominantly driven by the tangential temperature gradient do not occur. In this paper, we focus on the possibility of HTWs in droplets with moderate contact angles using numerical simulation. For this purpose, a three-dimensional numerical model is proposed, in which the deformation of the droplet surface induced by evaporation can be directly calculated from the local evaporation flux. We predict fan-like HTWs with three blades in the droplet. The waves are located at the center region of the droplet and are uniformly distributed throughout the circumference; they simultaneously propagate along the anticlockwise and radial directions. With evaporation, the blades become shorter, whereas the fan-like shape and wave number remain without significant variation. The behavior of the fluid flow inside the droplet and its oscillatory characteristics were analyzed carefully. The numerical results were verified by experiments, and confirm that HTWs would only occur under moderate contact angles. Additionally, a higher substrate temperature or a larger initial radius would promote the appearance of the irregular fan-like HTWs with two blades, whereas a sufficient small initial radius will inhibit the appearance of HTWs. The effects of the substrate temperature, the initial radius and the contact angle on the critical radius and the corresponding critical normal Marangoni number (Mav,c) for the disappearance of the HTWs were also studied.

Motion of an active particle in a linear concentration gradient

Janus particles self-propel by generating local tangential concentration gradients along their surface. These gradients are present in a layer whose thickness is small compared to the particle size. Chemical asymmetry along the surface is a prerequisite to generate tangential chemical gradients, which gives rise to diffusio-osmotic flows in a thin region around the particle. This results in an effective slip on the particle surface. This slip results in the observed “swimming” motion of a freely suspended particle even in the absence of externally imposed concentration gradients. Motivated by the chemotactic behavior of their biological counterparts (such as sperm cells, neutrophils, macrophages, bacteria, etc.), which sense and respond to external chemical gradients, the current work aims at developing a theoretical framework to study the motion of a Janus particle in an externally imposed linear concentration gradient. The external gradient along with the self-generated concentration gradient determines the swimming velocity and orientation of the particle. The dominance of each of these effects is characterized by a non-dimensional activity number A (ratio of applied gradient to self-generated gradient). The surface of Janus particle is modeled as having a different activity and mobility coefficient on the two halves. Using the Lorentz reciprocal theorem, an analytical expression for the rotational and translational velocity is obtained. The analytical framework helps us divide the parameter space of surface activity and mobility into four regions where the particle exhibits different trajectories.

Breakup process of liquid jet in gas film

In order to study the breakup process of liquid jet in gas film, the backlit photography technique and the VOF TO DPM method are used for experimental and simulation research respectively. Water and air are used as simulant media. Grid adaptive technology is used to refine the gas-liquid interface grid and improve the capture accuracy of the gas-liquid interface. The results show that there are two main breakup processes of liquid jet in gas film: column breakup and surface breakup. The local high-pressure zone in front of the liquid jet makes the jet have a large normal velocity gradient, which causes R-T instability. The surface wave that is generated by the R-T instability is mainly responsible for the liquid column breakup. When the thin liquid film reaches a column breakup point, the airflow penetrating the trough of the surface wave causes the jet column to break. The tangential velocity gradient is generated when the gas film bypasses the liquid jet surface, which causes K-H instability. The K-H surface waves cause ligaments and droplets to strip from the surface of the liquid jet. The local momentum ratio has an important influence on the breakup process of the liquid jet in gas film. When the local momentum ratio is low, the breakup of liquid jet is dominated by the K-H instability. As the local momentum ratio increases, the breakup of liquid jet is gradually dominated by R-T instability. The local momentum ratio plays an important role in the distribution range of the liquid jet in gas film. When the local momentum ratio is low, the ligaments and droplets caused by the liquid jet are mainly distributed within the range of gas film. As the local momentum ratio increases, part of the ligaments and droplets escape from the range of the gas film. The liquid jet penetrates the gas film when the local momentum ratio is greater than 0.74. The breakup length and the penetration depth are both affected by the local momentum ratio. The breakup length increases with the local momentum ratio increasing. The penetration depth also increases with the local momentum ratio, and the penetration depth increases significantly when the liquid jet penetrates the gas film.

A NEW IRIS LOCATION METHOD

A new iris location method is presented in this paper. When locating the iris inner boundary, a method for locating the iris inner boundary based on the variation coefficient and multiple disk-structural elements is proposed. The variation coefficient of the iris image is first calculated to be used for the image gray stretch, then the mathematical morphology operations with multiple disk-structural elements are applied to tackle the binary iris image, finally the connected region in line with geometric features of the circle is picked up and its center and radius are calculated, namely for obtaining the iris inner boundary. When positioning the iris outer boundary, a method for locating the iris outer boundary based on the variation coefficient and the gradient amplitude is presented. First, the Gaussian function is employed to filter the iris image using the variation coefficient as its standard deviation, then for each pixel on a circle which is centered at each neighborhood pixel near the circle enter of the iris inner boundary and has a radius with greater than that of the inner boundary, the radial gradient amplitude and tangential gradient amplitude of are computed, finally the iris outer boundary is determined on the maximum of the tangential gradient amplitude. The experiments reveal that this proposed method is effective, and has a fairly simple implementation, a low computational load, a fast location and good accuracy.

Central moment lattice Boltzmann method using a pressure-based formulation for multiphase flows at high density ratios and including effects of surface tension and Marangoni stresses

Simulation of multiphase flows, which are ubiquitous in nature and engineering applications, require coupled capturing or tracking of the interfaces in conjunction with the solution of fluid motion often occurring at multiple scales. In this contribution, we will present unified cascaded LB methods based on central moments for the solution of the incompressible two-phase flows at high density ratios and for capturing of the interfacial dynamics. Based on a modified continuous Boltzmann equation (MCBE) for two-phase flows, where a kinetic transformation to the distribution function involving the pressure field is introduced to reduce the associated numerical stiffness at high density gradients, a central moment cascaded LB formulation using multiple relaxation times for computing the fluid motion will be constructed. In this LB scheme, the collision step is prescribed by the relaxation of various central moments to their equilibria that are reformulated in terms of the pressure field obtained via matching to the continuous equilibria based on the transformed Maxwell distribution. Furthermore, the differential treatments for the effects of the source term representing the change due to the pressure field and of the source term due to the interfacial tension force and body forces appearing in the MCBE on different moments are consistently accounted for in this cascaded LB solver that computes the pressure and velocity fields. In addition, another cascaded LB scheme will be developed to solve for the interfacial dynamics represented by a phase field model based on the conservative Allen-Cahn equation that evolves interfaces by advection and under the competing effects due to a diffusion term and a phase segregation flux term. The latter is introduced into the cascaded LB scheme via a modification to the moment equilibria. Based on numerical simulations of a variety of two-phase flow benchmark problems at high density ratios and involving the effects of surface tension and its tangential gradients (Marangoni stresses), we will validate our unified cascaded LB approach and also demonstrate improvements in numerical stability.

Spontaneous thermocapillary motion of condensation droplets

The spontaneous movements of condensate droplets on either superhydrophobic surfaces or homogenous slippery surfaces are generally driven by capillary forces. It is difficult to shift a millidroplet without using, e.g., a wettability gradient or asymmetric bump. Its motion direction is not related to the surface temperature of the substrate, although the condensation strongly depends on the surface temperature. This Letter reports a self-excited thermocapillary motion during condensation on a heterogeneous slippery liquid-infused porous surface without an externally imposed tangential temperature gradient, where the droplet moves directionally toward cold areas on the surface. The spontaneous thermocapillary motion is driven by the thermocapillary force originating from the local nonuniform temperature distribution on the surface, which is several orders of magnitude larger than the capillary force for a millidroplet. Even a millidroplet could move on such a heterogeneous surface and move upward against the gravity on an inclined surface. In addition to the spontaneous motion directly related to the temperature of the cooling substrate, the dropwise condensation rate may be significantly increased up to two times compared to that of a homogenous slippery surface.

An implicit, sharp numerical treatment of viscous terms at arbitrarily shaped liquid-gas interfaces in evaporative flows

We present a novel methodology for accurate treatment of viscous terms in evaporation problems. The proposed scheme is an extension of the sharp viscous treatment of Kang et al. (2000) [7] to 3D phase change problems. To ensure accuracy and grid converging solutions, a new implicit approach to computing viscous fluxes across the phase interfaces is proposed, which was previously unavailable in fixed grid numerical schemes. Analytical relations were derived for the jump in velocity gradients across the 2D and 3D phase interfaces, an important constituent of the proposed scheme. The relations show a non-vanishing jump in the tangential gradients across the phase interface that are associated with evaporative flux and interfacial curvature. The proposed methodology demonstrated first order accuracy in canonical test cases. It is general and applicable to arbitrarily oriented interfaces, and can be readily implemented in existing evaporation flow solvers.

Numerical investigation on flow instability of sessile ethanol droplets evaporating in its pure vapor at low pressure

In order to understand the flow instability in the evaporation sessile droplet, a series of numerical simulations are carried out on the ethanol droplet evaporating in its pure vapor at low pressure. The contact radius of sessile droplet on the substrate is 2.5mm. The temperature of the ethanol vapor is fixed at 298K, and the corresponding saturated vapor pressure is 7615Pa. Results show that with the increase of the substrate temperature, different types of the thermal patterns appear in sequence. They are the steady axisymmetric pattern or multi-cell pattern driven by tangential temperature gradient, Bénard-Marangoni cells mainly induced by vertical temperature gradient, and longitudinal rolls caused by inclined temperature gradient. The relation between the evaporation rate and the contact angle is non-monotonic, which depends on the combined effects of Marangoni flow intensity, evaporating surface area as well as the length of heat transfer path from substrate to evaporation surface. The type of flow instability is independent on gravity. However, buoyancy convection changes the transition Marangoni number of the thermal patterns.

An immersed boundary formulation incorporating a two-layer wall model approach for RANS simulations with complex geometry

We propose an immersed boundary (IB) formulation incorporating a two-layer wall model approach for Reynolds-averaged Navier-Stokes (RANS) simulations with complex geometry. The main objective of the present study is to broaden the capability of the IB method to high-Reynolds number turbulent flows with non-equilibrium effects. A two-layer wall model is applied to the IB method by considering a thin boundary layer approximation near the wall in order to reduce the computational cost of resolving the turbulent boundary layer. Tangential pressure gradient and convective terms are included in the thin boundary layer equation to make the wall model suitable for non-equilibrium flows, in which cross-derivative terms are modeled along the wall-normal direction to maintain the ordinary differential form of the equation. Turbulent viscosity in the inner layer is determined using a simplified k−ϵ−fμ model, that performs better than the typical mixing-length model in the separation and reattachment regions, where the friction velocity uτ becomes zero. To impose the mass conservation constraint near the IB surface with marginal computational cost, the IB approximated domain method is applied. The proposed IB formulation is validated using numerical simulations of two-dimensional turbulent channel flow, backward-facing step flow, turbulent flow around a circular cylinder, and three-dimensional turbulent flow over a wall-mounted circular sphere with a relatively coarse grid near the IB surface. The results demonstrate that the proposed IB formulation captures the flow patterns well and accurately predicts wall shear stress for both equilibrium and non-equilibrium flows, even with a relatively coarse grid near the wall.

Evaluation of the Assumptions in the Steady-State Tropical Cyclone Self-Stratified Outflow Using Three-Dimensional Convection-Allowing Simulations

Abstract The criteria and assumptions that were used to derive the steady-state tropical cyclone intensity and structure theory of Emanuel and Rotunno are assessed using three-dimensional convection-allowing simulations using the Weather Research and Forecasting Model. One real-data case of Hurricane Patricia (2015) and two idealized simulations with and without vertical wind shear are examined. In all three simulations, the gradient wind balance is valid in the inner-core region above the boundary layer. The angular momentum M and saturation entropy surfaces s* near the top of the boundary layer, in the outflow region and along the angular momentum surface that passes the low-level radius of maximum wind MRMW are nearly congruent, satisfying the criterion of slantwise moist neutrality in the vicinity of MRMW. The theoretically derived maximum wind magnitude above the boundary layer compares well with the simulated maximum tangential wind and gradient wind using the azimuthally averaged pressure field during the intensification and quasi-steady state of the simulated storms. The Richardson number analysis of the simulated storms shows that small Richardson number (0 < Ri ≤1) exists in the outflow region, related to both large local shear and small static stability. This criticality of the Richardson number indicates the existence of small-scale turbulence in the outflow region. We also show that the stratification of temperature along the M surfaces at the outflow region for steady-state hurricanes is approximately applicable in these three-dimensional simulations, while the radial distribution of gradient wind is qualitatively comparable to the theoretical radial profiles. Some caveats regarding the theory are also discussed.

On the mechanism of symmetric vortex shedding

In this paper, we study the vortex-shedding mechanism of a square cylinder subjected to a mean flow with a superimposed pulsatile flow, under symmetric vortex shedding conditions. The near-body vortical events are interpreted in accordance with the two-dimensional vorticity transport equation, and the interactions between different vortical regions are quantified by circulation balance for a control volume in the near wake. Strong wake counter flow that splits into two branches, with one branch considerably disrupting the boundary layer while other branch cuts off supply of vorticity to the shear layer, is found to be an essential feature of symmetric vortex shedding. To further clarify the roles of correlated terms contributing to the vorticity generation and transport in the wake, the Reynolds number was varied while keeping other excitation parameters constant. This results in the intensity of symmetric vortex shedding increasing with Reynolds number. The increase in wake counter-flow strength with increased Reynolds number cannot be explained either by a relative velocity between the mean and pulsatile flow components or relative acceleration between bluff body and fluid. Based on the results it is inferred that the brief rolling up of the shear layer to form a wake vortex during a part of the pulsation cycle, together with the associated unsteady tangential pressure gradient around the body, contribute to the stronger wake counter flow at higher Reynolds numbers. At lower pulsation amplitudes, the tendency of the fluid in the wake to form strong counter flow can interact significantly with the transverse entrainment flow to alter the natural vortex-shedding mechanism, to give different vortex-shedding modes.

Iterative PIV interrogation for complex wall-bounded flows

The strategy of the iterative PIV algorithm is discussed with the aim of investigating transitional wall-bounded flows. It is proposed to start the algorithm with a coarse prediction of the large-scale tangential velocity gradient to eliminate its negative effect. A window correlation with image deformation technique is then iteratively conducted to capture the fluctuating flow structures. Synthetic particles are created based on the DNS result of a typical hypersonic transitional boundary layers to evaluate the performance of the algorithm. Effects of several parameters, including the mean tangential displacement gradient, particle diameter, particle density and background noise on the level on the evaluation, are systematically investigated. Four methods to initially predict the near-wall flow are compared, including the directly-used window correlation with image deformation (WC) (Scarano 2002 Meas. Sci. Technol. 13 R1–19), the WC method with static particles added in the solid region (Zhu et al 2013 Meas. Sci. Technol. 24 125302), the image gradiometry method (Nguyen et al 2009 Proc. 6th Int. Symp. on Turbulence and Shear Flow Phenomena (Seoul, Korea, 2009)) and the newly-proposed joint-translation-shear correlation method method. The results show that, compared with the other three approaches, the performance of the new method is greatly enhanced for evaluating the dynamical flow properties, even if the displacement gradient is very high. Finally, experimental data on artificially disturbed boundary layers from a low-speed and a high-Mach-number wind tunnel are used to evaluate the new method, which is able to evaluate the mean and instantaneous velocity fields in both cases.

Simulation of viscoelastic two-phase flows with insoluble surfactants

A stabilized finite element scheme is developed for computations of buoyancy driven 3D-axisymmetric viscoelastic two-phase flows with insoluble surfactants. The numerical scheme solves the Navier–Stokes equations for the fluid flow, Giesekus constitutive equation for the effects of viscoelasticity and simultaneously an evolution equation for the surfactant concentration on the interface. The interface is tracked by the coupled arbitrary Lagrangian–Eulerian (ALE) and Lagrangian approach. The interface-resolved moving meshes allow accurate incorporation of the interfacial tension force, Marangoni forces and the jumps in the material properties. Further, the tangential gradient operator technique is used to handle the curvature approximation in a semi-implicit manner. An one-level Local Projection Stabilization (LPS), which is based on an enriched approximation space and a discontinuous projection space, where both spaces are defined on a same mesh is used to stabilize the model equations. The stabilized numerical scheme allows us to use isoparametric second order conforming finite elements enriched with cubic bubble functions for velocity and viscoelastic stress, second order finite elements for surfactant concentration and discontinuous first order finite element for pressure. A number of computations are performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column with insoluble surfactants on the interface. The influence of the Marangoni number, initial surfactant concentration and Peclet number on the dynamics of the rising drop are analyzed. The numerical study shows that a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape without insoluble surfactants. The presence of insoluble surfactants forces the drop to rise slowly but the drop at the tail end is pulled up more. However, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape without insoluble surfactants. The presence of surfactants pulls the tail end of the drop up slightly and makes the tail flatter with/without small undulations depending on the magnitude of the surfactant concentrations.

A new least squares stabilized Nitsche method for cut isogeometric analysis

We derive a new stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions for elliptic problems of second order in cut isogeometric analysis (CutIGA). We consider C1 splines and stabilize the standard Nitsche method by adding a certain elementwise least squares terms in the vicinity of the Dirichlet boundary and an additional term on the boundary which involves the tangential gradient. We show coercivity with respect to the energy norm for functions in H2(Ω) and optimal order a priori error estimates in the energy and L2 norms. To obtain a well posed linear system of equations we combine our formulation with basis function removal which essentially eliminates basis functions with sufficiently small intersection with Ω. The upshot of the formulation is that only elementwise stabilization is added in contrast to standard procedures based on ghost penalty and related techniques and that the stabilization is consistent. In our numerical experiments we see that the method works remarkably well in even extreme cut situations using a Nitsche parameter of moderate size.

Residual Magnetic Field Non-Destructive Testing of Gantry Cranes.

Non-destructive tests of gantry cranes by means of the residual magnetic field (RMF) method were carried out for a duration of 7 years. Distributions of the residual magnetic field tangential and the normal components of their gradients were determined. A database of magnetograms was created. The results show that the gradients of tangential components can be used to identify and localize stress concentration zones in gantry crane beams. Special attention was given to the unsymmetrical distribution of the tangential component gradient on the surface of the crane beam No. 5 (which was the most loaded one). The anomaly was the effect of a slight torsional deflection of the beam as it was loaded. Numerical simulations with the finite element method (FEM) were used to explain this phenomenon. The displacement boundary conditions introduced into the simulations were established experimentally. Validation was carried out using the X-ray diffraction method, which confirmed the location of strain concentration zones (SCZs) identified by means of RMF testing.

Influence of Disc Tip Geometry on the Aerodynamic Performance and Flow Characteristics of Multichannel Tesla Turbines

As a competitive small-scale turbomachinery option, Tesla turbines have wide potential in various fields, such as renewable energy generation systems and small power equipment. This paper investigates the influence of disc tip geometry, including its profile and relative height, on the aerodynamic performance and flow characteristics of one-to-one and one-to-many multichannel Tesla turbines. The results indicate that compared to the turbine with blunt tips, the isentropic efficiency of the one-to-one turbine with sharp tips has a little decrease, which is because the relative tangential velocity gradient near the rotational disc walls decreases a little and additional vortices are generated at the rotor inlet, while that of the one-to-many turbine with sharp tips increases significantly, resulting from an increase in the relative tangential velocity in the disc channels and a decrease in the low Mach number and vortex area; for instance the turbine efficiency for the former relatively decreases by 3.6% and that for the latter increases by 13.5% at 30,000 r/min. In addition, the isentropic efficiency of the one-to-many turbine with sharp tips goes up with increasing relative height due to increasing improvement of flow status, and its increment rate slows down. A circular or elliptic tip performs better with lower relative height and a triangular tip behaves better with higher relative height. To sum up, a blunt disc tip is recommended for the one-to-one turbine, and a sharp disc tip is for the one-to-many turbine. The relative height and tip profile of the one-to-many turbine should be determined according to their effects on turbine performance, manufacturing difficulty and mechanical deformation.