Solid-body Rotation Research Articles

Acceleration of superrotation in simulated hot Jupiter atmospheres

Context. Atmospheric superrotating flows at the equator are a nearly ubiquitous result when conducting simulations of hot Jupiters. One theory explaining how this zonally-coherent flow reaches equilibrium has already been developed in the literature. This understanding, however, relies on the existence of either an initial superrotating flow or a sheared flow, coupled with a slow evolution that permits a linear steady state to be reached. Aims. A consistent physical understanding of superrotation is needed for arbitrary drag and radiative timescales, along with the relevance of taking linear steady states into account, needs to be assessed. Methods. We obtained an analytical expression for the structure, frequency, and decay rate of propagating waves in hot Jupiter atmospheres around a state at rest in the 2D shallow-water β-plane limit. We solved this expression numerically and confirmed the robustness of our results with a 3D linear wave algorithm. We then compared it with 3D simulations of hot Jupiter atmospheres and studied the nonlinear momentum fluxes. Results. We show that under strong day-night heating, the dynamics do not transit through a linear steady state when starting from an initial atmosphere in solid body rotation. We further demonstrate that nonlinear effects favor the initial spin-up of superrotation and that acceleration due to the vertical component of the eddy-momentum flux is critical to the initial development of superrotation. Conclusions. We describe the initial phases of the acceleration of superrotation, including the consideration of differing radiative and drag timescales, and we conclude that eddy-momentum-driven superrotating equatorial jets are robust, physical phenomena in simulations of hot Jupiter atmospheres.

On the stages of vortex decay in an impulsively stopped, rotating cylinder

The flow within an infinitely long cylinder exhibiting solid-body rotation (SBR) is impulsively stopped. The complete decay of the initial SBR is captured by means of direct numerical simulations for a wide range of Reynolds numbers ( $Re$ ). Five distinct stages are identified during the decay process according to their flow structure and their underlying mechanisms of kinetic-energy dissipation. Initially, the laminar boundary layer undergoes a primary centrifugal instability, which causes the formation of coherent Taylor rolls. The flow then becomes turbulent, once the Taylor rolls are corrupted by secondary instabilities. Within the turbulent stage, two phases are distinguished. In the first turbulent phase, the SBR core is still intact and turbulence is sustained. The mean velocity profile is well described by the superposition of a near-wall region, a retracting SBR core and an intermediate region of constant angular momentum. In the latter region, the magnitude of angular momentum in viscous units $l^{+}(Re)$ is approximately constant in time. In the second turbulent phase, the SBR core breaks down, turbulence starts to decay exponentially and the kinetic energy of the mean flow decays logarithmically. Eventually, the flow relaminarises and the velocity profile of the analytical solution for purely laminar decay is recovered, albeit at an earlier temporal instant due to the net effect of transition and turbulence.

Turbulence structures over realistic and synthetic wall roughness in open channel flow at Reτ = 1000

ABSTRACTTurbulence structures in flow over three types of wall roughness: sand-grain, cube roughness and a realistic, multi-scale turbine-blade roughness, are compared to structures observed in flow over a smooth wall in open channel flow at , using direct numerical simulations. Two-point velocity correlations, length scales, inclination angles, and velocity spectra are analysed, and the applicability of Townsend's outer layer similarity hypothesis [Townsend. The structure of turbulent shear flow. Cambridge: Cambridge University Press; 1976] to these parameters was examined. Results from linear stochastic estimation suggest that, near the wall, the quasi-streamwise vortices observed in smooth-wall flow are present in the large-scale recessed regions of multi-scale roughness, whereas they are replaced by a pair of ‘head-up, head-down’ horseshoe structures in sandgrain and cube roughness, similar to those observed by Talapatra and Katz [Coherent structures in the inner part of a rough-wall channel flow resolved using holographic PIV. J Fluid Mech. 2012;711:161–170]. The configuration of conditional eddies near the wall suggests that the kinematical process of vortices differ for each kind of rough surface. Eddies over multiscale roughness are conjectured to obey a growth mechanism similar to those over smooth walls, while around the cube roughness the head-down horse-shoe vortices of Talapatra and Katz [Coherent structures in the inner part of a rough-wall channel flow resolved using holographic PIV. J Fluid Mech. 2012;711:161–170] may undergo solid-body rotation on top of a cube roughness on account of the strong shear layer, shortening the longitudinal extent of near-wall structure and promoting turbulence production during this process. These results illustrate the sensitivity of turbulence structure to the roughness texture, particularly within the roughness sublayer.

Simulation of a shut-down transient in the Francis-99 turbine model

Due to the new intermittent electric energy sources, hydropower is forced to run more and more at off-design conditions and to regulate the operating conditions. This causes flow instabilities with pressure fluctuations, and load variations that may deteriorate the machine. Previous numerical research on the flow in water turbines has mainly focused on steady operating conditions at BEP and off-design. More knowledge is needed on transients between operating conditions, and start-up and shut-down procedures. This requires dynamic meshes that both rotate the runner, using a coupling interface to the non-rotating part of the mesh, and morph the guide vane mesh due to the change in guide vane angle during the transient. Special care needs to be taken to apply boundary conditions that resemble the experimental condition to be used for validation. The present work addresses a shut-down transient of the Francis-99 turbine (https://www.ntnu.edu/nvks/francis-99), using OpenFOAM-2.3.x. Turbulence is modelled by LES based on the dynamic one-equation eddy viscosity sub-grid model and the cube root volume filter width, although the temporal and spatial resolution requirements have not been assessed in these preliminary results. The inlet and outlet boundary conditions are set using pressure conditions. The outlet static pressure is taken from the experiment at BEP. The inlet total pressure is adjusted to yield the same flow rate as in the experiment at BEP. The flow rate is given by the solution. The runner region of the mesh rotates with a solid-body rotation, while the guide vane region of the mesh is morphing due to the continuous change in guide vane angle. Rotating and stationary parts of the mesh are coupled with an arbitrary mesh interface (AMI). The results show that the methodology can be used to capture the experimentally observed flow features during transients.

Numerical investigation on the transport equation in spherical coordinates via generalized moving least squares and moving kriging least squares approximations

The main aim of this paper is to present new and simple numerical methods for solving the time-dependent transport equation on the sphere in spherical coordinates. We use two techniques, namely generalized moving least squares and moving kriging least squares to find the new formulations for approximating the advection operator in spherical coordinates such that any singularities have been omitted. Another feature of the methods considered here is that they do not depend on the background mesh or triangulation for approximation, which they could be applied on the transport equation in spherical coordinates easily with different distribution points. Furthermore, due to the eigenvalue stability of the dicretized advection operator via two proposed approximations, an implicit-explicit linear multistep method has been applied to discretize the time variable. The fully discrete scheme obtained here yields a linear system of algebraic equations at each time step, which is solved via the biconjugate gradient stabilized algorithm with zero-fill incomplete lower-upper (ILU) preconditioner. Three well-known test problems, namely “solid body rotation”, “vortex roll-up” and “deformational flow” are solved to demonstrate our developments. Also for the first test problem, we apply a simple positivity-preserving filter at the end of each time step, which keeps the transported variable positive.

Droplets. II. Internal Velocity Structures and Potential Rotational Motions in Pressure-dominated Coherent Structures

Abstract We present an analysis of the internal velocity structures of the newly identified sub-0.1 pc coherent structures, droplets, in L1688 and B18. By fitting 2D linear velocity fields to the observed maps of velocity centroids, we determine the magnitudes of linear velocity gradients and examine the potential rotational motions that could lead to the observed velocity gradients. The results show that the droplets follow the same power-law relation between the velocity gradient and size found for larger-scale dense cores. Assuming that rotational motion giving rise to the observed velocity gradient in each core is a solid-body rotation of a rotating body with a uniform density, we derive the “net rotational motions” of the droplets. We find a ratio between rotational and gravitational energies, β, of ∼0.046 for the droplets, and when including both droplets and larger-scale dense cores, we find β ∼ 0.039. We then examine the alignment between the velocity gradient and the major axis of each droplet, using methods adapted from the histogram of relative orientations introduced by Soler et al. We find no definitive correlation between the directions of velocity gradients and the elongations of the cores. Lastly, we discuss physical processes other than rotation that may give rise to the observed velocity field.

The Formation and Evolution of Turbulent Swirling Vortex Rings Generated by Axial Swirlers

The present work investigates the formation process and early stage evolution of turbulent swirling vortex rings, by using planar Particle Image Velocimetry (PIV) and Large Eddy Simulation (LES). Vortex rings are produced in a piston-nozzle arrangement with swirl generated by 3D-printed axial swirlers in experiments. Idealised solid-body rotation is applied in LES to evaluate the effect of nozzle exit velocity profile in experiments. The Reynolds number (Re) based on the nozzle diameter D and the slug velocity U0 in the nozzle is 20,000. The swirl number S generated ranges from 0 (zero-swirl vortex ring) and 1.1, covering the two critical swirl numbers previously identified in a swirling jet. Both PIV and LES results show that the formation number F decreases linearly as S increases, with the maximum F ≈ 2.6 at S = 0 (produced by the swirler with straight vanes) and minimum F = 1.9 at S = 1.1. The corresponding maximum attainable circulation in the nozzle axis parallel plane also diminishes with increasing S. Evolution of compact rings produced by a stroke ratio L/D = 1.5 reveals that circulation decay rate is largely proportional to S. The trajectory of the vortex core in the axial direction, hence the ring axial propagation velocity, decreases as S, while that in the radial direction and the radial propagation velocity, increase with S. An empirical scaling function is proposed to scale these variables.

Observational investigation of the 2013 near-Earth encounter by asteroid (367943) Duende

On 15 February 2013, the asteroid 367943 Duende (2012 DA14) experienced a near-Earth encounter at an altitude of 27,700 km or 4.2 Earth radii. We present here the results of an extensive, multi-observatory campaign designed to probe for spectral and/or rotational changes to Duende due to gravitational interactions with the Earth during the flyby. Our spectral data reveal no changes within systematic uncertainties. Post-flyby lightcurve photometry places strong constraints on the rotation state of Duende, showing that it is in non-principal axis rotation with fundamental periods of P1 = 8.71 ± 0.03 and P2 = 23.7 ± 0.2 h. Multiple lightcurve analysis techniques, coupled with theoretical considerations and delay-Doppler radar imaging, allow us to assign these periods to specific rotational axes of the body. In particular we suggest that Duende is now in a non-principal, short axis mode rotation state with a precessional period equal to P1 and oscillation about the symmetry axis at a rate equal to P2. Temporal and signal-to-noise limitations inherent to the pre-flyby photometric dataset make it difficult to definitively diagnose whether these periods represent a change imparted due to gravitational torques during the flyby. However, based on multiple analysis techniques and a number of plausibility arguments, we suggest that Duende experienced a rotational change during the planetary encounter with an increase in its precessional rotation period. Our preferred interpretation of the available data is that the precession rate increased from 8.4 h prior to the flyby to 8.7 h afterwards. A companion paper by Benson et al. (2019) provides a more detailed dynamical analysis of this event and compares the data to synthetic lightcurves computed from a simple shape model of Duende. The interpretation and results presented in these two works are consistent with one another. The ultimate outcome of this campaign suggests that the analytic tools we employed are sufficient to extract detailed information about solid-body rotation states given data of high enough quality and temporal sampling. As current and future discovery surveys find more near-Earth asteroids, the opportunities to monitor for physical changes during planetary encounters will increase.

Waves of large amplitudes in geophysical and technical applications

The paper addresses the asymptotic models that explicitly describe the structure of solitary waves with recirculation zones for some special but important incident stream configurations, which occur in geophysical and industrial applications . The key feature that allows building a theoretical model is that shear and/or stratification of the media in geophysical applications or axisymmetric swirling incoming flows in technical applications have an almost linear equation for vorticity. It happens when the upstream flow is almost linearly stratified in the case of geophysical applications and a swirling flow has almost solid body rotation if industrial applications are considered. It is shown that the resulting equation for the stationary amplitude function of the solitary waves is described by a generalized Korteweg-de Vries equation in the outer zone, consistenty matched with a new complicated equation in the inner zone, which contains the recirculation core. This recirculation zone exists for wave amplitudes that are slightly greater than a certain critical amplitude, for which a wave breaking initially occurs. The recirculation zones have a universal structure, so that their width increases up to the semi-infinite bore as the amplitude of the wave increases from a critical amplitude to a certain maximum amplitude. This amplitude may be sensitive to the actual configuration of the incident flow. Regardless of the physical nature, solitary waves are described by two types of equations with different non-linearities. The type of nonlinearity in the inner zone where the vortex core is located is then caused by the position of the vortex core at the outer boundary or in the middle of the flow.

Mapping the zonal structure of Titan's northern polar vortex

Saturn exhibits an obliquity of 26.7° such that the largest moon, Titan, experiences seasonal variations including the formation of a polar vortex in the winter hemisphere. Titan's polar vortex is characterised by cold stratospheric temperatures due to the lack of insolation over the winter pole, and an increase in trace gas abundance as a result of complex organic chemistry in the upper atmosphere combined with polar subsidence. Meridional variations in temperature and gas abundance across the vortex have previously been investigated, but there has not yet been any in-depth study of the zonal variations in the temperature or composition of the northern vortex. Here we present the first comprehensive two-dimensional seasonal mapping of Titan's northern winter vortex. Using 18 nadir mapping sequences observed by the Composite InfraRed Spectrometer (CIRS) instrument on-board Cassini, we investigate the evolution of the vortex over almost half a Titan year, from late winter through to mid summer (Ls = 326 − 86°, 2007–2017). We find the stratospheric symmetry axis to be tilted from the solid body rotation axis by around 3.5°, although our results for the azimuthal orientation of the tilt are inconclusive. We find that the northern vortex appears to remain zonally uniform in both temperature and composition at all times. A comparison with vortices observed on Earth, Mars, and Venus shows that large-scale wave mechanisms that are important on other terrestrial planets are not as significant in Titan's atmosphere. This allows the northern vortex to be more symmetrical and persist longer throughout the annual cycle compared to other terrestrial planets.

Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence

In this paper, we present an experimental investigation of the turbulent saturation of the flow driven by the parametric resonance of inertial waves in a rotating fluid. In our set-up, a half-metre wide ellipsoid filled with water is brought to solid-body rotation, and then undergoes sustained harmonic modulation of its rotation rate. This triggers the exponential growth of a pair of inertial waves via a mechanism called the libration-driven elliptical instability. Once the saturation of this instability is reached, we observe a turbulent state for which energy is injected into the resonant inertial waves only. Depending on the amplitude of the rotation rate modulation, two different saturation states are observed. At large forcing amplitudes, the saturation flow mainly consists of a steady, geostrophic anticyclone. Its amplitude vanishes as the forcing amplitude is decreased while remaining above the threshold of the elliptical instability. Below this secondary transition, the saturation flow is a superposition of inertial waves which are in weakly nonlinear resonant interaction, a state that could asymptotically lead to inertial wave turbulence. In addition to being a first experimental observation of a wave-dominated saturation in unstable rotating flows, the present study is also an experimental confirmation of the model of Le Reun et al. (Phys. Rev. Lett., vol. 119 (3), 2017, 034502) who introduced the possibility of these two turbulent regimes. The transition between these two regimes and their relevance to geophysical applications are finally discussed.

Thermoelectric convection in a dielectric liquid inside a cylindrical annulus with a solid-body rotation

We study thermoelectric convection in a cylindrical annulus of dielectric liquid under rotation and a fixed temperature gradient. Convection onset is delayed and stationary helical vortices become oscillatory columnar vortices for a rotation rate above a value dependent on the radius ratio.

Dynamical controls on the longevity of a non-linear vortex : The case of the Lofoten Basin Eddy

The Lofoten Basin is the largest oceanic reservoir of heat in the Nordic Seas, and the site of important heat fluxes to the atmosphere. An intense permanent anticyclone in the basin impacts the regional hydrography, energetics, and ecosystem. Repeated sampling of this Lofoten Basin Eddy from dedicated cruises, autonomous profiling gliders, and acoustically-tracked subsurface floats enables the documentation of its dynamics and energetics over the course of 15 months. The eddy core, in nearly solid-body rotation, exhibits an unusually low vertical vorticity close to the local inertial frequency and important strain rates at the periphery. Subsurface floats as deep as 800 m are trapped within the core for their entire deployment duration (up to 15 months). The potential vorticity is reduced in the core by two orders of magnitude relative to the surroundings, creating a barrier. In the winter, this barrier weakens and lateral exchanges and heat flux between the eddy and the surroundings increase, apparently the result of dynamical instabilities and a possible eddy merger. Based on a simple energy budget, the dissipation timescale for the eddy energy is three years, during which wintertime convection seasonally modulates potential and kinetic energy.

Transient flows and reorientations of large-scale convection in a cubic cell

The transient processes of a turbulent large-scale convective circulation (LSC) in a cubic cell are investigated using large-eddy simulations for Rayleigh number Ra = 108 and Prandtl number Pr = 0.7. For the first time, we have explicitly shown that LSC is accompanied by large-scale azimuthal flows with non-zero total angular momentum. It is also shown that solid-body rotation of the entire fluid is not realized. It is found that correlation between rotation of LSC plane and the mean azimuthal motion is high during quasiperiodic oscillations of LSC near the diagonal plane and relatively weak during LSC reorientations. We propose a new plausible scenario for the reorientations of the LSC in a cube that does not involve a mean azimuthal flow. Instead of a single-roll, we introduce the superposition of a pair of large-scale orthogonal quasi-two-dimensional (Q2D) rolls and the reorientation of the LSC occurs as a result of the cessation of one of the Q2D rolls. This scenario is consistent with all known experimental and numerical data.

The Specific Angular Momentum Radial Profile in Dense Cores: Improved Initial Conditions for Disk Formation

The determination of the specific angular momentum radial profile, $j(r)$, in the early stages of star formation is crucial to constrain star and circumstellar disk formation theories. The specific angular momentum is directly related to the largest Keplerian disk possible, and it could constrain the angular momentum removal mechanism. We determine $j(r)$ towards two Class 0 objects and a first hydrostatic core candidate in the Perseus cloud, which is consistent across all three sources and well fit with a single power-law relation between 800 and 10,000\,au: $j_{fit}(r)=10^{-3.60\pm0.15}\left(r/\textrm{1,000au}\right)^{1.80\pm0.04}$ km s$^{-1}$ pc. This power-law relation is in between solid body rotation ($\propto r^2$) and pure turbulence ($\propto r^{1.5}$). This strongly suggests that even at 1,000\,au, the influence of the dense core's initial level of turbulence or the connection between core and the molecular cloud is still present. The specific angular momentum at 10,000\,au is $\approx3\times$ higher than previously estimated, while at 1,000\,au it is lower by $2\times$. We do not find a region of conserved specific angular momentum, although it could still be present at a smaller radius. We estimate an upper limit to the largest Keplerian disk radius of 60\,au, which is small but consistent with published upper limits. Finally, these results suggest that more realistic initial conditions for numerical simulations of disk formation are needed. Some possible solutions include: a) use a larger simulation box to include some level of driven turbulence or connection to the parental cloud, or b) incorporate the observed $j(r)$ to setup the dense core kinematics initial conditions.

Stencil selection algorithms for WENO schemes on unstructured meshes

In this paper, a family of stencil selection algorithms is presented for WENO schemes on unstructured meshes. The associated freedom of stencil selection for unstructured meshes, in the context of WENO schemes present a plethora of various stencil selection algorithms. The particular focus of this paper is to assess the performance of various stencil selection algorithm, investigate the parameters that dictate their robustness, accuracy and computational efficiency. Ultimately, efficient and robust stencils are pursued that can provide significant savings in computational performance, while retaining the non-oscillatory character of WENO schemes. This is achieved when making the stencil selection algorithms adaptive, based on the quality of the cells for unstructured meshes, that can in turn reduce the computational cost of WENO schemes. For assessing the performance of the developed algorithms well established test problems are employed. These include the least square approximation of polynomial functions, linear advection equation of smooth functions and solid body rotation test problem. Euler and Navier-Stokes equations test problems are also pursued such as the Shu-Osher test problem, the Double Mach Reflection, the supersonic Forward Facing step, the Kelvin-Helmholtz instability, the Taylor-Green Vortex, and the flow past a transonic circular cylinder.

Derivatives on the downstream side of a moving, curved shock

Novel relations are developed for the tangential and normal derivatives of the pressure, density, and velocity components just downstream of a regular point on a curved shock wave. The perfect gas flow is three dimensional with a nonuniform freestream and the flow may be unsteady. The analysis starts with data in a fixed, laboratory-frame Cartesian coordinate system. By means of an orthogonal transformation, a coordinate system is introduced via a solid-body translation and two solid-body rotations. This Cartesian system is shock based and limits the analysis to a selected shock point at a given instant of time when the flow is unsteady. The analysis is a local one in that Taylor series expansions are utilized for the configuration of the shock and the upstream gradients of the velocity, pressure, and density. The coefficients in the last three expansions must satisfy Euler equation constraints. These expansions are performed in both Cartesian systems, with occasional preference given to the laboratory-frame expansions where there are known data. The derivatives are in terms of a third coordinate system that is shock based and utilizes the flow plane at the selected shock point. Exact, explicit results are provided for the tangential and normal derivatives in as simple a form as possible. In the Taylor series expansion version for the tangential derivatives, the relative importance of various factors can be assessed. These are the upstream velocity, pressure, and density gradients, curvatures, and shock-shape terms not associated with curvatures. A different type of discussion is provided for the more involved normal derivatives.

Nozzle exit conditions and the heat transfer in non-swirling and weakly swirling turbulent impinging jets

Investigations have been conducted into turbulent impinging jets but the exact flow dynamics and mechanisms leading to the observed heat transfer distributions at the impingement plane remain outstanding. In particular, use of different swirl generators (vanes, twisted inserts) means the role of varying inflow conditions (at the nozzle exit plane x/D = 0) should be studied to resolve its role on the observed convective heat transfer trends. The present paper studies axisymmetric turbulent weakly swirling (S = 0.31) jets (D = 40 mm) impinging onto a heated plate. Parameters varied include inflow conditions and the effects of impingement distance (H/D = 2, 4, and 6). The Reynolds Averaged Navier Stokes (RANS) equations are used to model the jets using the k-kl-ω turbulence model, which is benchmarked against other models. Three azimuthal ( ) velocity profiles at a Reynolds (Re) number of 24,600 are used at the nozzle exit plane: Uniform (UP), Solid Body Rotation (SBR), and Parabolic Profiles (PP). The start of the wall jet region, designated through elevated levels of turbulent kinetic energy correlates well with the widely observed first peak in Nu distribution. This is however extremely sensitive to the imposition of any swirl, with the application of even weak swirl (S = 0.31) minimally modifying flow dynamics (in the upstream jet region) and leading to recirculation zones stabilized at the impingement plane. This occurs in near-field impingement (H/D) for some inflow conditions (S031-UP), but not others thereby highlighting the significance varied nozzle and swirl generation methods on trends observed in the literature. The imposition of elevated levels of turbulence at the nozzle inflow (x/D = 0) appreciably modifies the heat transfer distribution, particularly in far-field impingement (H/D = 6).

Stencil selection algorithms for WENO schemes on unstructured meshes

In this paper, a family of stencil selection algorithms is presented for WENO schemes on unstructured meshes. The associated freedom of stencil selection for unstructured meshes, in the context of WENO schemes present a plethora of various stencil selection algorithms. The particular focus of this paper is to assess the performance of various stencil selection algorithm, investigate the parameters that dictate their robustness, accuracy and computational efficiency. Ultimately, efficient and robust stencils are pursued that can provide significant savings in computational performance, while retaining the non-oscillatory character of WENO schemes. This is achieved when making the stencil selection algorithms adaptive, based on the quality of the cells for unstructured meshes, that can in turn reduce the computational cost of WENO schemes. For assessing the performance of the developed algorithms well established test problems are employed. These include the least square approximation of polynomial functions, linear advection equation of smooth functions and solid body rotation test problem. Euler and Navier-Stokes equations test problems are also pursued such as the Shu-Osher test problem, the Double Mach Reflection, the supersonic Forward Facing step, the Kelvin-Helmholtz instability, the Taylor-Green Vortex, and the flow past a transonic circular cylinder.

Performance analysis of multi-stage gravitational water vortex turbine

In this study, the performance of a multi-stage gravitational water vortex turbine (GWVT), which is assembled in a conical basin, is investigated analytically and experimentally. Each runner of the multi-stage GWVT is independent in terms of power generation through telescopic shaft arrangement. Performance parameters, such as rotational speed, torque, power and efficiency, are analyzed under various load conditions in which the effects of design parameters, such as rotors ratio, offset distance between neighbouring runners, and intra-staging and inter-staging of two-stage and three-stage GWVT, are considered. In the multi-stage GWVT, the profile of the blades of upstream runners result in minimal vortex distortion. Thus, the power generation capacity of downstream runners are ultimately enhanced because the performance of latter runners strongly depend on the head utilization capacity of the former runner. Furthermore, the runners with blades tilted on a vertical plane are best suited for the position near the bottom of the basin, whereas cross-flow blades are recommended at the top position. Moreover, the present study suggests that the rotors ratio of the neighbouring runners be selected in such a way that the two runners have the same rotor-to-basin diameter ratio with optimum offset distance. Multi-staging in GWVT demonstrates the development of a combined effect of solid body rotation and free vortex. It also shows a prominent improvement in the overall performance of the turbine compared with that of single-stage GWVT. The proposed mathematical model successfully predicts the experimental results qualitatively and quantitatively, thereby proving its reliability and robustness.