Pressure Gradient Case Research Articles

Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel

The unsteady flow of viscoelastic fluid with the fractional derivative Maxwell model (FDMM) in a channel is studied in this note. The exact solutions are obtained for an arbitrary pressure gradient by means of the finite Fourier cosine transform and the Laplace transform. Two special cases of pressure gradient are discussed. Some results given by the classical models with integer-order are included in this note.

Numerical prediction of transition boundary-layer flows using new intermittency transport equation

AbstractIn this paper, a new transport equation for the intermittency factor is proposed to model the transition flows. The intermittency behaviour of the transition flows is incorporated into the differential methods for solving the boundary-layer equations, which deal numerically with the basic partial differential equations. The present model accuracy and validity have been tested against a series of recent published experiments, for low Reynolds number, including flows with different freestream turbulence intensities and different pressure-gradients, such as aerofoil and flat plate flows. A comparison of the present method and two different prediction techniques is also given.The significance of the proposed transport intermittency equation is to reproduce the streamwise variation of the intermittency factor in the transition zone. This method is found suitable and reliable to predict flows with positive or favourable pressure-gradient cases and with turbulence intensity level up to 6%.The method also confirmed the importance of estimating the start of transition, present formula. The present formula is suitable and straightforward to use. For all test cases good agreement between the computed results and the experimental data are observed.

Wake flow in adverse pressure gradient

In the interest of improving the predictability of high-lift systems at maximum lift conditions, a series of fundamental experiments were conducted to study the effects of adverse pressure gradient on a wake flow. Mean and fluctuating velocities were measured with a two-component laser-Doppler velocimeter. Data were obtained for several cases of adverse pressure gradient, producing flows ranging from no reversed flow to massively reversed flow. While the turbulent Reynolds stresses increase with increasing size of the reversed flow region, the gradient of Reynolds stress (− ∂ uv/ ∂y) does not. Computations using various turbulence models were unable to reproduce the reversed flow.

The Dynamics of Northwest Summer Winds over the Santa Barbara Channel

A mesoscale model is used to examine the dynamics of northwest flow over the Santa Barbara Channel region. Three cases are considered, each characterized by typical summertime synoptic conditions, but with differences in pressure gradient strength and marine boundary layer depth (MBL). The first case examines a relatively deep MBL and strong pressure gradient. Case 2 is characterized by a more shallow MBL and weaker pressure gradient, and case 3 represents a transition from a deep MBL to shallow conditions. In all cases, simulated surface winds show reasonable agreement with observations over most of the model domain, with the exception of regions near abrupt terrain changes. Results from the model indicate that the flow with a deep MBL (∼400 m) and strong pressure gradient (case 1) is supercritical, causing regions of acceleration and expansion in the lee of Point Conception. When the MBL is shallow (∼150 m) (case 2), a transcritical flow scenario exists with subcritical flow upstream from Point Conception and a supercritical flow region over the Santa Barbara Channel and downstream from the Channel Islands. Flow over the channel is strongly affected by diurnal heating in shallow MBL cases, reversing direction in step with a land breeze circulation induced by nighttime cooling. The land breeze forces an internal wave disturbance that propagates westward across the channel, eliminating the supercritical flow region in the lee of Point Conception. Conditions with a deep MBL (∼400 m) produce less variability in the surface winds, except for the region sheltered by the Santa Ynez Mountains. An expansion fan is still evident in this case, but it is produced by the interaction of the flow with higher terrain north and east of the channel. The low hills on Point Conception and the Channel Islands do not have a large blocking effect on the surface flow when the MBL is deep. Analysis of the momentum budget supports the conclusion that the boundary layer behaves like a transcritical hydraulic flow when the MBL is shallow. Except for the open ocean region, the Coriolis term is minor in comparison with the pressure and advection terms. Diurnal heating effects are evident in the nearshore pressure term, which varies from offshore during the late evening to onshore in the afternoon. These effects are most significant when the MBL is shallow and can augment the hydraulically forced pressure pattern, causing a stronger expansion fan in the late afternoon and a collapse of the expansion fan during the early morning.

Unsteady laminar hydromagnetic fluid–particle flow and heat transfer in channels and circular pipes

Abstract The problem of unsteady laminar flow and heat transfer of a particulate suspension in an electrically conducting fluid through channels and circular pipes in the presence of a uniform transverse magnetic field is formulated using a two-phase continuum model. Two different applied pressure gradient (oscillating and ramp) cases are considered. The general governing equations of motions (which include such effects as particulate phase stresses, magnetic force, and finite particle-phase volume fraction) are non-dimensionalized and solved in closed form in terms of Fourier cosine and Bessel functions and the energy equations for both phases are solved numerically since they are non-linear and are difficult to solve analytically. Numerical solutions based on the finite-difference methodology are obtained and graphical results for the fluid-phase volumetric flow rate, the particle-phase volumetric flow rate, the fluid-phase skin-friction coefficient and the particle-phase skin-friction coefficient as well as the wall heat transfer for plane and axisymmetric flows are presented and discussed. In addition, these numerical results are validated by favorable comparisons with the closed-form solutions. A comprehensive parametric study is performed to show the effects of the Hartmann magnetic number, the particle loading, the viscosity ratio, and the temperature inverse Stokes number on the solutions.

Predictive model of the turbulent flow of dilute gas–particulate suspension in a vertical pipe

A Chapman–Enskog closure approximation for the third order fluctuating velocity correlations in the particle phase of a turbulent dilute gas–particulate suspension is used to formulate the closed system of equations and boundary conditions for fully developed flow of dilute but densely loaded suspension of “high-inertia” particles in a vertical pipe. The particle size and concentration are assumed to be sufficiently small so that direct interparticle interactions (e.g., collisions) can be neglected, and the main mechanism inducing particle velocity fluctuations is the interaction between particles and turbulent flow. The case of moderate gas pressure gradient is studied thus modeling a number of practical applications (e.g., riser flow). The resulting set of continuum equations, consisting of mass and momentum conservation and Reynolds stress equations for the particulate phase, is free of empirical parameters. Although in a general case the effective stress in the particulate phase is anisotropic, the criterion is obtained showing that in a wide range of parameters typical for applications, this anisotropy may be neglected so that the equations for the individual diagonal components of the Reynolds stress tensor may be reduced to just one conservation equation for the particle fluctuation energy. The numerical solution shows the bifurcation of flow properties at a certain gas pressure gradient, thus providing an explicit criterion (i.e., a critical pressure gradient for the given total mass flux of solid particles) for upward particulate flow. The profiles of particle volume fraction and velocity are calculated, the former demonstrating the phenomenon of particle segregation toward the wall. A generalization of the model for the nearly developed flow is discussed, and an estimate is derived for the vertical distance required for the flow to become fully developed.A Chapman–Enskog closure approximation for the third order fluctuating velocity correlations in the particle phase of a turbulent dilute gas–particulate suspension is used to formulate the closed system of equations and boundary conditions for fully developed flow of dilute but densely loaded suspension of “high-inertia” particles in a vertical pipe. The particle size and concentration are assumed to be sufficiently small so that direct interparticle interactions (e.g., collisions) can be neglected, and the main mechanism inducing particle velocity fluctuations is the interaction between particles and turbulent flow. The case of moderate gas pressure gradient is studied thus modeling a number of practical applications (e.g., riser flow). The resulting set of continuum equations, consisting of mass and momentum conservation and Reynolds stress equations for the particulate phase, is free of empirical parameters. Although in a general case the effective stress in the particulate phase is anisotropic, the cr...

Modeling of Flow Transition Using an Intermittency Transport Equation

A new transport equation for intermittency factor is proposed to model transitional flows. The intermittent behavior of the transitional flows is incorporated into the computations by modifying the eddy viscosity, μt, obtainable from a turbulence model, with the intermittency factor, γ:μt*=γμt. In this paper, Menter’s SST model is employed to compute μt and other turbulent quantities. The proposed intermittency transport equation can be considered as a blending of two models—Steelant and Dick and Cho and Chung. The former was proposed for near-wall flows and was designed to reproduce the streamwise variation of the intermittency factor in the transition zone following Dhawan and Narasimha correlation and the latter was proposed for free shear flows and a realistic cross-stream variation of the intermittency profile was reproduced. The new model was used to predict the T3 series experiments assembled by Savill including flows with different freestream turbulence intensities and two pressure-gradient cases. For all test cases good agreements between the computed results and the experimental data were observed. [S0098-2202(00)02302-6]

Microbubble Skin Friction Reduction under the Influence of Pressure Gradients and Curved Surface

The influence of streamwise pressure gradients and curved surfaces on microbubble skin friction reduction was examined in cavitation tunnel at free-stream speeds of 5, 7 and 10m/sec. A 1.5m long, 0.3m width flat plate and 2D convex model were used. In order to change streamwise pressure gradient, using temporary slopping bottom changed sectional area at the test section.The effect of skin friction reduction under favorable pressure gradients tended to rather reduce compared with the zero pressure gradient case. In opposition, the adverse pressure gradients tended to rather increase. Moreover, there was a condition that the skin friction reduction effect increased extremely under the adverse pressure gradient. Air bubbles under that condition looked like cloud and the void fraction of inner layer was high. There seems to be possible to find an effective microbubble method by the optimization with air bubble diameter and distribution.

A New Model for Free-Stream Turbulence Effects on Boundary Layers

A model has been developed to incorporate more of the physics of free-stream turbulence effects into boundary layer calculations. The transport in the boundary layer is modeled using three terms: (1) the molecular viscosity, ν; (2) the turbulent eddy viscosity, εT, as used in existing turbulence models; and (3) a new free-stream-induced eddy viscosity, εf. The three terms are added to give an effective total viscosity. The free-stream-induced viscosity is modeled algebraically with guidance from experimental data. It scales on the rms fluctuating velocity in the free stream, the distance from the wall, and the boundary layer thickness. The model assumes a direct tie between boundary layer and free-stream fluctuations, and a distinctly different mechanism than the diffusion of turbulence from the free-stream to the boundary layer assumed in existing higher order turbulence models. The new model can be used in combination with any existing turbulence model. It is tested here in conjunction with a simple mixing length model and a parabolic boundary layer solver. Comparisons to experimental data are presented for flows with free-stream turbulence intensities ranging from 1 to 8 percent and for both zero and nonzero streamwise pressure gradient cases. Comparisons are good. Enhanced heat transfer in higher turbulence cases is correctly predicted. The effect of the free-stream turbulence on mean velocity and temperature profiles is also well predicted. In turbulent flow, the log region in the inner part of the boundary layer is preserved, while the wake is suppressed. The new model provides a simple and effective improvement for boundary layer prediction.

Linear and nonlinear development of localized disturbances in zero and adverse pressure gradient boundary-layers

Localized disturbances in laminar boundary-layers at Reynolds number Reδ*=950 were studied using direct numerical simulation. Instability mechanisms in both an adverse and zero pressure gradient were investigated by introducing three different three-dimensional disturbances. The first disturbance was centered around a pair of oblique waves in Fourier space, the second around a plane wave, while the third was axisymmetric. For small amplitudes, the first disturbance developed into a wave-packet of oblique waves in adverse pressure gradient and into a streaky structure with a trailing wave-packet in a zero pressure gradient. The second disturbance developed into a wave-packet centered around plane waves in both pressure gradients. The third disturbance developed into a wave-packet of plane waves in adverse pressure gradient and, due to the transient growth mechanism, into a streaky structure in a zero pressure gradient. For finite-amplitude plane wave-packets in a zero pressure gradient, a subharmonic secondary instability was observed which subsequently developed into an elongated Λ structure. The secondary instability was less significant in adverse pressure gradient due to the large growth rate of the primary instability. Breakdown was observed as high-frequency oscillations on the spike over the head of the Λ vortices. Providing the initial amplitude was sufficiently large, the vortex pair yielded the fastest route to turbulence. The main growth mechanism in this scenario, in addition to the exponential growth in the adverse pressure gradient case, was the nonlinear excitation of transient growth of streaks by interacting oblique modes. Disturbances dominated by streaks needed substantially larger amplitudes before secondary instability or breakdown occurred. In that case, breakdown was shifted from the spike to an instability of the rear part of the streaks.

Direct numerical simulation of self-similar turbulent boundary layers in adverse pressure gradients

Direct numerical simulations of the Navier–Stokes equations have been carried out with the objective of studying turbulent boundary layers in adverse pressure gradients. The boundary layer flows concerned are of the equilibrium type which makes the analysis simpler and the results can be compared with earlier experiments and simulations. This type of turbulent boundary layers also permits an analysis of the equation of motion to predict separation. The linear analysis based on the assumption of asymptotically high Reynolds number gives results that are not applicable to finite Reynolds number flows. A different non-linear approach is presented to obtain a useful relation between the freestream variation and other mean flow parameters. Comparison of turbulent statistics from the zero pressure gradient case and two adverse pressure gradient cases shows the development of an outer peak in the turbulent energy in agreement with experiment. The turbulent flows have also been investigated using a differential Reynolds stress model. Profiles for velocity and turbulence quantities obtained from the direct numerical simulations were used as initial data. The initial transients in the model predictions vanished rapidly. The model predictions are compared with the direct simulations and low Reynolds number effects are investigated.

Heat transfer from rough, protruding discretely heated strips with and without flow acceleration

Experimental heat transfer results on the effects of rough, protruding, discretely heated strips are presented. Data are presented for two-dimensional (2-D), distributed heat sources for three cases—flush-mounted smooth, protruding smooth, and protruding rough—under zero pressure gradient and accelerated turbulent boundary-layer conditions. For the zero pressure-gradient cases, experiments are performed on repeated, protruding discretely heated strips using surface roughness as a heat transfer enhancement technique. The accelerated cases are conducted to investigate the existence and strength of a synergistic interaction between boundary-layer flow acceleration and surface roughness as a heat transfer enhancement mechanism. For the zero pressure-gradient cases, it is found that under the conditions of these experiments, significant heat transfer enhancements (10–100%) can be obtained with the rough surface. For the smooth-flush arrangement with acceleration, the Stanton numbers decreased by up to 16% with increasing acceleration strength. For the protruding smooth arrangement, the Stanton numbers remain almost constant over all acceleration strengths. For the protruding rough arrangement, the Stanton numbers increase by up to 7% with increasing acceleration strength.

Asymptotic theory of two-phase gas-solid flow through a vertical tube at moderate pressure gradient

Based on the equations, constitutive relations and boundary conditions of the kinetic theory of colliding particles in a gas-solid suspension, the approximate theory of the steady, developed vertical flow of a gas-particulate mixture is developed for the case of moderate gas pressure gradient in a vertical tube. The basic equations and boundary conditions show a singular behaviour of the solution of the problem at the wall. The method of matched asymptotic expansions is applied to develop a boundary layer-type theory for the flow parameters of the particulate phase. The basic equations in the bulk flow are reduced to a system of two ordinary integrodifferential equations for the particle-phase concentration and mean kinetic energy of particle velocity fluctuations (particle-phase pseudotemperature). The distributions of the particle concentration and velocity are found in both the bulk and the boundary layer. The solutions shows the bifurcation of flow parameters, and an explicit criterion is derived to identify a range of the given macroscopic parameters corresponding to upward or downward particulate flow. The integrated parameters (total fluxes of the gas and particle phase) are calculated.

Turbulence measurements in a Mach 2.9 boundary layer including mild pressure gradients

Mass-weighted-time, or Favre-averaged, turbulence measurements were acquired in a Mach 2.9 air boundary layer for three pressure gradient cases. Curved walls were used to produce the pressure gradient and streamline curvature effects. The boundary layers along a curved wall expansion (β -0.5 and Re θ = 14.5 x 10 3 ), the expansion region (β -1.2 and Re θ = 11.5 x 10 3 ) following an initial compression (β 1.0), and a nominally zero pressure gradient (Re θ = 10.9 x 10 3 ) were investigated. Laser Doppler velocimetry and cross-film anemometry were used to acquire direct and detailed surveys of the Reynolds shear stress (ρu'v') and the mean transverse Favre velocity fluctuation (v). Velocity profiles were also acquired.

Transport in mixed ionic electronic conductor solid oxide membranes with porous electrodes: Large pressure gradient

We extend the model of a mixed ionic electronic conductor (MIEC) oxygen perm-selective solid oxide membrane with porous electrodes, previously treated in the limit of small pressure gradient, to the case of arbitrary pressure gradient. The model is based on the assumption that the oxygen vacancy concentration varies only weakly with gas pressure and that the surface exchange reaction rate varies as √ p where p is the oxygen gas pressure. While the model applies specifically to SrCo 0.8Fe 0.2O 3 − δ it is expected to be applicable also to other mixed ionic electronic conductor materials with high acceptor dopant concentrations. Numerical solution of the diffusion-reaction equations, in the case of a large pressure gradient, show that the enhancement in oxygen permeability due to the porous electrodes scales with the simple analytical expression for the flux enhancement obtained in the limit of a small pressure gradient. Enhancement of over two orders of magnitude in oxygen flux are predicted for membranes with large surface area porous electrodes.

The Effects of Adverse Pressure Gradients on Momentum and Thermal Structures in Transitional Boundary Layers: Part 1—Mean Quantities

The effects of adverse pressure gradients on the thermal and momentum characteristics of a heated transitional boundary layer were investigated with free-stream turbulence ranging from 0.3 to 0.6 percent. The acceleration parameter, K, was kept constant along the test section. Both surface heat transfer and boundary layer measurements were conducted. The boundary layer measurements were conducted with a three-wire probe (two velocity wires and one temperature wire) for two representative cases, K1 = −0.51 × 10−6 and K2 = −1.05 × 10−6. The surface heat transfer measurements were conducted for K values ranging from −0.045 × 10−6 to −1.44 × 10−6 over five divergent wall angles. The Stanton numbers of the cases with adverse pressure gradients were greater than that of the zero-pressure-gradient turbulent correlation in the low-Reynolds-number turbulent flow, and the difference increased as the adverse pressure gradient was increased. The adverse pressure gradient caused earlier transition onset and shorter transition length based on Rex, Reδ*, and Reθ in comparison to zero-pressure-gradient conditions. As expected, there was a reduction in skin friction as the adverse pressure gradient increased. In the U+−Y+ coordinates, the adverse pressure gradients had a significant effect on the mean velocity profiles in the near-wall region for the late-laminar and early transition stations. The mean temperature profile was observed to precede the velocity profile in starting and ending the transition process, opposite to what occurred in favorable pressure gradient cases in previous studies. A curve fit of the turbulent temperature profile in the log-linear region for the K2 case gave a conduction layer thickness of Y+ = 9.8 and an average Prt = 0.71. In addition, the wake region of the turbulent mean temperature profile was significantly suppressed.

The Effects of Adverse Pressure Gradients on Momentum and Thermal Structures in Transitional Boundary Layers: Part 2—Fluctuation Quantities

The effects of adverse pressure gradients on the thermal and momentum characteristics of a heated transitional boundary layer were investigated with free-stream turbulence ranging from 0.3 to 0.6 percent. Boundary layer measurements were conducted for two constant-K cases, K1 = −0.51 × 10−6 and K2 = −1.05 × 10−6. The fluctuation quantities, u′, ν′, t′, the Reynolds shear stress (uν), and the Reynolds heat fluxes (νt and ut) were measured. In general, u′/U∞, ν′/U∞, and νt have higher values across the boundary layer for the adverse pressure-gradient cases than they do for the baseline case (K = 0). The development of ν′ for the adverse pressure gradients was more actively involved than that of the baseline. In the early transition region, the Reynolds shear stress distribution for the K2 case showed a near-wall region of high-turbulent shear generated at Y+ = 7. At stations farther downstream, this near-wall shear reduced in magnitude, while a second region of high-turbulent shear developed at Y+ = 70. For the baseline case, however, the maximum turbulent shear in the transition region was generated at Y+ = 70, and no near-wall high-shear region was seen. Stronger adverse pressure gradients appear to produce more uniform and higher t′ in the near-wall region (Y+ < 20) in both transitional and turbulent boundary layers. The instantaneous velocity signals did not show any clear turbulent/nonturbulent demarcations in the transition region. Increasingly stronger adverse pressure gradients seemed to produce large non turbulent unsteadiness (or instability waves) at a similar magnitude as the turbulent fluctuations such that the production of turbulent spots was obscured. The turbulent spots could not be identified visually or through conventional conditional-sampling schemes. In addition, the streamwise evolution of eddy viscosity, turbulent thermal diffusivity, and Prt, are also presented.

Resonant growth of three-dimensional modes in Falkner–Skan boundary layers with adverse pressure gradients

This work documents the spatial development of a triad of instability waves consisting of a plane TS mode and a pair of oblique modes with equal-opposite wave angles which are undergoing subharmonic transition in Falkner-Skan boundary layers with adverse pressure gradients. The motivation for this study is that for wings with zero or moderate sweep angles, transition is most likely to occur in the adverse pressure gradient region past the maximum thickness point and, starting with low initial amplitudes, subharmonic mode transition is expected to be the predominant mechanism for the first growth of of three-dimensional modes. The experiment follows that of Corke & Mangano (1989) in which the disturbances to produce the triad of waves are introduced by a spanwise array of heating wires located near Branch I. The initial conditions are carefully controlled. These include the initial amplitudes, frequencies, relative phase and oblique wave angles. The basic flow consisted of a Falkner-Skan (Hartree) boundary layer with a dimensionless pressure gradient parameter in the range -0.06 ≤ β H ≤ -0.09. The frequency of the TS wave was selected to be near the most amplified based on linear theory. The frequency of the oblique waves was the subharmonic of the TS frequency. The oblique wave angles were set to give the largest secondary growth ( 60°). Compared to similar conditions in a Blasius boundary layer, the adverse pressure gradient was observed to lead to an extra rapid growth of the two- and three-dimensional modes. In this there was a relatively larger maximum amplitude of the fundamental mode and considerably shortened amplitude saturation region compared to zero pressure gradient cases. Analysis of these results includes frequency spectra, the wall-normal distributions of each mode amplitude, and mean velocity profiles. Finally, the streamwise amplitude development is compared with the amplitude model from the nonlinear critical layer analysis of Goldstein & Lee (1992).

Supersonic flux-split procedure for second moments of turbulence

A finite volume procedure which combines flux-difference splitting and flux-vector splitting is presented in this paper. The diffusive fluxes are treated explicitly to preserve the upwinding structure of the split Euler fluxes. This procedure is extended in this paper from its essentially laminar or eddy viscosity form to include the equations for the six components of the Reynolds stress tensor and an additional equation for the solenoidal dissipation. The models used for the unclosed terms are presented as are the extensions of the numerical procedures to cover the turbulence equations. To validate the proposed procedures, a compressible turbulent flow over a flat plate at Mach number M∞ = 2.87 and Reynolds number per unit length Re/m of 6.5X10 7 was calculated and compared with experimental measurements. Also simulated was the case when the foregoing boundary-layer flow was made to pass over a ramp. A strong adverse pressure gradient case in which boundary-layer thickness-to-curvature ratio (δ/R c ) is 0.1 was considered. In this case, the supersonic turbulent boundary layer experiences the combined effects of an adverse pressure gradient, bulk compression, and concave streamline curvature.

Unsteady axial viscoelastic pipe flows

The main objective of this work is to examine in detail basic unsteady pipe flows and to investigate any new physical phenomena. We take the viscoelastic upper-convected Maxwell fluid as our non-Newtonian model and consider the flow of such a fluid in pipes of uniform circular cross-section in the following three cases: 1. (a) when the pressure gradient varies exponentially with time; 2. (b) when the pressure gradient is pulsating; 3. (c) a starting flow under a constant pressure gradient. In the first problem we looked separately at the pressure gradient rising exponentially with time and falling exponentially with time, i.e. the pressure gradient is proportional to e ± α 2 τ . The behaviour of the flow field depends to a large extent on β where β 2 = α 2(1 ± Hα 2) with H being the quotient of the Weissenberg and Reynolds numbers. In both cases for small | βη|, η being the radial distance from the axis, the velocity profiles are seen to be parabolic. However, for large | βη| the flows are vastly different. In the case of increasing pressure gradient the flow depicts boundary-layer characteristics while for decreasing pressure gradient the velocity depends on the wall distance. The case of a pulsating pressure gradient is investigated in the second problem. Here the pressure gradient is proportional to cos nτ. Again the flow depends to a large extent on a parameter β ( β 2 = i n − n 2 H). For small values of | βη| the velocity profile is parabolic. However, it is found that, unlike Newtonian fluids, the velocity distribution for the upper-convected Maxwell fluid is not in phase with the exciting pressure distribution. In the case of large | βη| the solution displays a boundary-layer characteristic and the phase of the motion far from the wall is shifted by half a period. The final problem examines a flow that is initially at rest and then set in motion by a constant pressure gradient. A closed form solution has been obtained with the aid of a Fourier-Bessel series. The variation of the velocity across the pipe has been sketched and comparison made with the classical solution.