Thermal Boundary Layer Profiles Research Articles

Radiation effect on MHD three-dimensional stagnation-point flow comprising water-based graphene oxide nanofluid induced by a nonuniform heat source/sink over a horizontal plane surface

This research aims to study the 3D magnetohydrodynamics stagnation-point flow (SPF) over a horizontal plane surface (HPS) carrying water-based graphene oxide (GO) nanoparticles caused by an irregular heat source/sink used in heat transfer procedures. In addition, a Tiwari–Das model is used to inspect the dynamics of fluid flow behavior and heat transmission features of the nanoparticles with experiencing the impacts of thermal radiation. The acquired nonlinear set of partial differential equations (PDEs) is transfigured to a system of ordinary differential equations (ODEs) using similarity transformations. The accumulative dimensionless ODEs are then further tackled in MATLAB using the bvp4c solver. Tables and figures are prepared for the execution of several relevant constraints along with nodal/saddle indicative parameter, internal heat source/sink parameter, radiation parameter and nanoparticles volume fraction which divulges and clarify more accurately the posited quantitative data and graphical findings. Also, the velocity profile decelerated in the axial and transverse coordinate axes for a higher value of the nanoparticle volume fraction but the dimensionless temperature distribution is augmented. Additionally, thermal boundary layer thickness and profile of temperature enriches with higher impressions of radiation constraint. However, the internal heat sink factor declines the profiles of temperature while escalating with the superior value of the internal heat source parameter.

Method for Determining Adiabatic Film Effectiveness in Presence of Thermal Boundary Layer

Abstract In this paper, we present a new method for determining adiabatic film effectiveness in film cooling experiments with nonuniform inlet temperature distributions, in particular the situation of an inlet thermal boundary layer. This might arise in a quasi-steady experiment due to loss of heat from the mainstream flow to the inlet contraction walls, for example. In this situation, the thermal boundary layer would be time-varying. Adiabatic film effectiveness is generally normalized by the difference between mainstream and coolant gas temperatures. Most importantly, these temperatures are generally assumed to be spatially—and, possibly temporally—uniform at the system inlet. In experiments with nonuniform inlet temperature, the relevant hot gas temperature for a particular point of interest on a surface is not easily determined, being a complex function of both the inlet temperature profile and the flow field between the inlet and the point of interest. In this situation, adiabatic film effectiveness cannot be uniquely defined using conventional processing techniques. We solve this problem by introducing the concept of equivalent mainstream effectiveness, a nondimensional temperature for the mainstream that can be used to represent the thermal boundary layer profile at the inlet plane, or the effective temperature of the mainstream gas—which we refer to as the equivalent mainstream temperature—entrained into the mixing layer affecting the wall temperature at a particular point of interest. By using data from two or more time instants during an experiment, we simultaneously solve for equivalent mainstream effectiveness and true adiabatic film effectiveness, that is, the adiabatic film effectiveness that we would measure in an experiment with both steady and uniform inlet temperature. This is an important transformation because the true adiabatic effectiveness has a clear physical interpretation and is a more transportable quantity between systems (comparisons between different experiments, between experiment and computational fluid dynamics (CFD), etc.). The proposed method is experimentally demonstrated using full-surface infrared (IR) thermography measurements of a cooled rotor blade platform, operated in a transonic linear cascade facility at matched engine conditions. Film effectiveness measurements processed in the conventional way suffer from inlet thermal boundary layer effects, rendering them both time-varying (as heat is released to the walls of the tunnel) and, in certain regions, nonphysical in magnitude. The proposed technique renders the same data insensitive to time, and everywhere within physically reasonable limits. By demonstrating independence to the particular (time-varying) inlet temperature profile, we demonstrate the advantages of the proposed technique.

Magneto-nanofluid coolants past heated shrinking/stretching surfaces: Dual solutions and stability analysis

Recent advancement in nanotechnology has provides a veritable platform for the emergence of a better ultrahigh-performance coolant known as nanofluid for many engineering and industrial technologies. In this study, we examine the influence of magnetic field on heat transfer enhancement of nanofluids coolants consisting of Cu-water, Al2O3-water, and Fe3O4-water over a slippery but convectively heated shrinking and stretching surfaces. Based on some realistic assumptions, the nonlinear model differential equations are obtained and numerically tackled using shooting procedure with Runge-Kutta-Fehlberg integration scheme. The existence dual solutions in the specific range of shrinking surface parameters is found. Temporal stability analysis to small disturbances is performed on these dual solutions. It is detected that the upper solution branch is stable, substantially realistic with positive smallest eigenvalues while the lower solution branch is unstable with negative smallest eigenvalues. Influence of numerous emerging parameters on the momentum and thermal boundary layer profiles, skin friction, Nusselt number and heat transfer enhancement rate are depicted graphically and quantitatively discussed.

On Behavioral Response of Microstructural Slip on the Development of Magnetohydrodynamic Micropolar Boundary Layer Flow

In this paper, the concept of microstructural slip is introduced in the flow of micropolar fluids pertinent to model various physical situations. The flow is modeled by a set of PDEs which are transformed to a nonlinear system of ODEs by employing boundary layer transformations. The system of governing equations is implemented using MATLAB bvp4c function along with the initial-boundary conditions. The code is validated by comparing the computed results in the limiting case with the available literature. Influence of microstructural slip on the skin friction coefficient and Nusselt number along with hydrodynamic and thermal boundary layer profiles is studied and discussed. It is found that, in the presence of microstructural slip, the microrotational velocity boundary layer thickness decreases up to a maximum of 37.5% in its value, in comparison to the case where there is no microstructural slip effect. The results predict that, in the presence of first-order translational slip, the microrotations have shown counterrotational phenomena in comparison to the case where there is no translational slip effect. Moreover, second-order translational slip results in declining the microrotational velocity and associated layer thickness.

LBM simulation of piezo fan in square enclosure

PurposeThis paper aims to investigate the use of a piezo fan in an enclosure on wall heat transfer and thermal boundary layer profile in constant wall temperature situation.Design/methodology/approachThe governing partial differential equations of mass, momentum and energy in addition to boundary conditions are solved by lattice Boltzmann method. The problem is solved numerically using D2Q9 population's model and Bhatnagar–Gross–Krook collision model with a code written in MATLAB.FindingsThe effects of Prandtl number (Pr) and the frequency of piezo fan vibrations are critically investigated on the hydrothermal characteristics of the square cavity. The mesh independency study and the validation of the proposed model are accomplished with numerical results of Ghia et al. (1982) and analytical solution of pure conduction very good agreement is found between present results and benchmark findings. Generally, with increasing beam frequency, the heat removal from heat source increased. It is found that, for all Prandtl numbers, wall Nusselt number will increase with the increase of the beam frequency. This enhancement is more intense in higher Prandtl number.Originality/valueBased on these results, the use of piezo fan in an enclosure can be classified as standalone as well as heat sink integrated cooling solution.

Experiments on flow and heat transfer characteristics of a rectangular channel with a built-in adiabatic square cylinder

Experimental investigation of unsteady flow past a square cylinder confined in a channel and the associated heat transfer phenomena in laminar flow regime is carried out using purely non-intrusive diagnostic technique. A Mach-Zehnder interferometer has been employed to make the path-averaged measurements of the quantities of interest. Five different Reynolds numbers are considered and the heat transfer rates in the channel with and without the presence of square cylinder have been compared in the wake region of the built-in cylinder. Vortex shedding frequencies are determined using schlieren technique. Both the temperature field and the vortex shedding frequencies have been completely determined using non-intrusive techniques. The whole field experimental results depict significant augmentation in the wall heat transfer rates in the presence of a cylinder when compared with its counterpart (channel without cylinder). Flow instabilities induced by the square cylinder alter the thermal boundary layer profile and hence temperature gradients near the channel walls become steeper and consequently the wall heat transfer is significantly increased.

Experimental study of heat transfer performance of compact wavy channel with nanofluids as the coolant medium: Real time non-intrusive measurements

Heat transfer performance of nanofluids in the context of corrugated compact channels has been experimentally investigated through real-time, non-intrusive diagnostic technique. The corrugated channel (dimensions: 50 mm (length) × 10 mm (width) × 4 mm (depth); hydraulic diameter of 5.7 mm) comprises of two horizontal wavy plates aligned in an in-phase configuration. Amplitude and period of the wavy plates have been kept constant. Experiments have been performed with varying concentrations of Al2O3/water-based nanofluids for Re = 350–1000. For reference, experiments have also been conducted using water as the base fluid and a plane channel with the same hydraulic diameter as that of the corrugated channel. Convective field has been mapped using a Mach-Zehnder interferometer. Results have been qualitatively as well as quantitatively interpreted to characterize the heat transfer performance of nanofluids as the coolant medium. Interferograms recorded under infinite fringe setting mode captured the phenomena of flow detachment and re-attachment at the wavy sections of the corrugated channel and their dependence on nanofluids concentration. Thermal boundary layer profiles at such sections are seen to be locally perturbed due to flow detachment and its subsequent attachment to the principal wall of the channel. Quantitative data retrieved through the wedge fringe setting interferograms revealed a clear enhancement in heat transfer coefficient for the combination of nanofluids and corrugated channel in comparison with that achieved through water in the plane and/or corrugated channel. For any given Re and channel configuration, the heat transfer coefficient is seen to increase with increasing concentration of dilute nanofluids. For the wavy channel, an enhancement of ≈37% in heat transfer rates with 0.02% volume concentration of nanofluids is achieved at Re = 350 in comparison to the plane channel at the similar levels of nanoparticles volume concentration.

Non-intrusive investigation of flow and heat transfer characteristics of a channel with a built-in circular cylinder

Interferometry-based experimental investigation of heat transfer phenomena associated with a channel fitted with a circular cylinder has been reported. Experiments have been performed with water as the working fluid, and the range of Reynolds number considered is 75 ≤ Re ≤ 165. The circular cylinder, placed at the inlet section of the channel, provides a blockage ratio of 0.5. The experimental methodology has been benchmarked against the results of transient numerical simulations. In order to assess the performance of the channel fitted with a circular cylinder for possible heat transfer enhancement from the channel wall(s), experiments have also been performed on a plane channel (without a cylinder). The interferometry-based experiments clearly highlighted the influence of the built-in cylinder in generating the flow instabilities and alterations in the thermal boundary layer profile along the heated wall of the channel. The phenomenon of vortex shedding behind the cylinder was successfully captured. A gradual increase in the vortex shedding frequency was observed with increasing Reynolds number. Quantitative data in the form of two-dimensional temperature distributions revealed an increase in the strength of wall thermal gradients in the wake region of the cylinder due to the periodic shedding of the vortices. In turn, a clear enhancement in the wall heat transfer rates was observed for the case of the channel fitted with a cylinder vis-à-vis the plane channel. To the best of the knowledge of the authors, the work reported is one of the first attempts to provide the planar field experimental data for a channel configuration with a built-in circular cylinder using non-intrusive imaging techniques and has the potential to serve as one of the benchmark studies for validating the existing as well as future numerical studies in the related area.

Numerical Modeling of Transpiration-Cooled Turbulent Channel Flow with Comparisons to Experimental Data

Transpiration cooling is a promising active cooling technique that provides coolant through a porous wall at a rate sufficient to maintain temperatures below a desired limit. The present study uses ANSYS® CFX to numerically investigate the influence of cooling gas injection on the temperature, velocity, and wall skin friction in the boundary layer of a subsonic turbulent hot-gas channel flow for various blowing ratios (), hot-gas temperatures (), and Mach numbers (, ). Here, a cooled permeable porous ceramic matrix composite sample with a hot-gas channel flow is simulated using a monolithic and two-domain approach. For the two-domain approach, separate solvers are used for the hot-gas and porous domains, respectively. These are applied alternatively and coupled to each other by boundary conditions imposed at the respective interface. The simulations are validated against experimental data and provide complementary insight into the effects of the cooling, which cannot be assessed from experimental measurements alone. The results show that, although a monolithic approach would be more convenient from a numerical point of view, a two-domain approach has proven to be superior in matching experimental data for this study. Furthermore, improved thermal boundary-layer profiles downstream of the porous sample are obtained by including the aft wall in the coupling process. This study adds to the limited theoretical and numerical investigations found in the literature on transpiration cooling.

Periodic momentum and thermal boundary layer mixed convection flow around the surface of a sphere in the presence of viscous dissipation

Numerical solutions for the periodic laminar boundary layer mixed convection flow around the surface of a heated sphere in the presence of viscous dissipation have been obtained by solving the governing equations using an implicit finite difference numerical technique. The fluid under consideration is assumed to be viscous and incompressible. Periodic momentum and thermal boundary layer profiles for different positions of x around the surface of the sphere are evaluated. The features of the obtained results for different values of mixed convection parameter λ, Prandtl number Pr, viscous dissipation parameter N, and frequency parameter ω are shown graphically. The obtained results confirm significant effect of all these mentioned parameters on periodic momentum and thermal boundary layer mixed convection flow around different positions of the sphere.

Nonintrusive Diagnostics of Nanofluids-Based Natural Convection Heat Transfer over a Heated Cylinder

The phenomena of nanofluids-based natural convection heat transfer around a heated cylinder have been studied using nonintrusive real-time diagnostics. Experiments have been performed with a horizontally placed heated cylinder immersed in bulk ambient conditions with water and alumina oxide/water-based dilute nanofluids as the working fluids. Volumetric concentrations of alumina nanoparticles have been varied from 0.005 to 0.02%. A classical Mach–Zehnder interferometer has been used to record the line-of-sight projection data of the convective field around the heated cylinder. The strength of thermal gradients prevailing around the surface of the heated cylinder was seen to be clearly affected by the variations in the nanofluid concentration, thus indicating the plausible role of nanoparticles in disrupting the thermal boundary-layer profiles around the cylinder surface. Significant enhancement in the heat transfer coefficient was observed at the highest levels of concentration of the dilute nanofluids employed, whereas the spatially averaged Nusselt numbers around the cylinder surface showed a marginal deterioration at the lower range of volumetric concentrations. With a further increase in nanoparticles concentration, the Nusselt numbers showed an increasing trend, although not as significant as observed in the case of the heat transfer coefficient. Plausible mechanisms responsible for these observations have been delved upon using purely nonintrusive whole field measurements.

Performance evaluation of compact channels with surface modifications for heat transfer enhancement: An interferometric study in developing flow regime

Performance evaluation of surface roughened compact channels for heat transfer applications has been investigated using non-intrusive, real time laser-based interferometric technique with water as the coolant medium. The lower wall of the channel has been roughened by creating hemispherical inward dimples. Projection data of the temperature field has been recorded using a Mach Zehnder interferometer. In order to facilitate direct comparison, experiments have also been conducted in smooth channel of similar dimensions. Results have been presented in the form of thermal boundary layer profiles, whole field temperature distributions and local variations of heat transfer coefficients. Direct interferometric measurements clearly reveal the disruption of thermal boundary layer due to the presence of inward dimples. Near wall temperature gradients were seen to be stronger in the case of dimpled channel in comparison with that of the smooth one resulting into a clear enhancement in heat transfer rates. At low Reynolds numbers, variation of heat transfer coefficients along the length of the dimpled channel showed the presence of local maxima. On the other hand, the corresponding profiles for the smooth channels showed a monotonic decrease with respect to the axial direction. The dynamic measurements, that are purely non-intrusive, revealed an improved thermal performance of surface roughened compact channels.

Exploring the potential of nanofluids for heat transfer augmentation in dimpled compact channels: Non-intrusive measurements

Potential of the combination of nanofluids and dimpled channels for possible heat transfer enhancement has recently been demonstrated by Li et al. (2014) [36] in the context of small length scale channels through numerical simulations. The present work reports real time, non-intrusive experimental investigation of this concept wherein the convective heat transfer enhancement has been achieved with the coupled effects of Al2O3/water-based dilute nanofluids and surface roughened compact channels. One of the thermally active walls of the channel has been engraved with hemispherical inward dimples of 3mm diameter with a pitch of 8mm. Forced convection experiments have been conducted with water and Al2O3/water-based dilute nanofluids as the coolant medium for a range of Reynolds numbers. Real time measurements have been made in a completely non-intrusive manner using a Mach Zehnder interferometer and the images of the convective fields have been recorded in infinite as well as wedge fringe setting modes of the interferometer. Results revealed a clear influence of the combination of nanoparticles and surface protrusions on phenomena such as disruption of thermal boundary layer profiles, periodic flow separation and re-attachments along the principal wall of the channel. The coupled effects of inward dimples and nanoparticles resulted into a clear reduction in the thickness as well as flattening of the thermal boundary layer in comparison with that achieved with the base fluid. An enhancement by ≈66.5% in the average heat transfer coefficient with 0.05% volumetric concentration of alumina nanoparticles was achieved with respect to that of the de-ionized water at a Reynolds number of Re=350.

Non-intrusive dynamic measurements of nanofluid-based heat transfer phenomena under thermally developing flow regime in the context of compact channels

Laser interferometry-based forced convective heat transfer experiments in the context of compact channels using nanofluids are reported. Experiments have been carried out under thermally-developing flow conditions for a range of Reynolds numbers. De-ionized water and alumina/water-based dilute nanofluids have been employed as the coolant medium. The nanofluid-based experiments employ three volumetric concentrations of 0.01%, 0.03% and 0.05%. Projection data of the convective fields have been recorded using a Mach–Zehnder interferometer. Phenomena such as thermal boundary layer profiles and thickness of thermal boundary layer have been interpreted based on the infinite fringe setting mode of the interferometer, while the quantitative information in the form of whole-field temperature distributions and heat transfer coefficients have been retrieved from the wedge fringe setting. Effects of increasing Reynolds numbers and volumetric concentrations of nanofluids on thermal boundary layer profiles and the resultant heat transfer rates have been investigated. Results of the experiments carried out at low Reynolds number (Re<513) reveal that the thermal boundary layer profiles are nearly insensitive to the changes in the concentrations of dilute nanofluids, while a significant enhancement in the heat transfer coefficient is seen as the concentrations of nanofluid is increased. On other hand, at higher Reynolds numbers, increasing concentrations of dilute nanofluids leads to a considerable increase in the thickness of thermal boundary layer that is accompanied with a mild deterioration in the heat transfer rates. Based on the primary findings of the experiments, that are purely non-intrusive in nature, various plausible mechanisms that govern the heat transfer characteristics of dilute nanofluids have been discussed.

Direct numerical simulation of the compression stroke under engine-relevant conditions: Evolution of the velocity and thermal boundary layers

The unsteady velocity and thermal boundary layers during the compression stroke under engine-relevant conditions are investigated using direct numerical simulation (DNS). The initial conditions are derived by a precursor DNS of the intake stroke, avoiding the use of any artificial initial and boundary conditions. The results show decreasing velocity and thermal boundary-layer thicknesses towards Top Dead Center (TDC) as a result of the decreasing kinematic viscosity. Compared to the fluctuating flow field, the azimuthally-averaged velocities are found to have a minor effect on the boundary-layer profiles, and as a result the boundary-layer structures at all engine walls during compression are similar. The averaged velocity and thermal boundary-layer profiles deviate strongly from the law of the wall, which is the basis for many wall heat transfer models in internal combustion engines. In density wall-normal units, the averaged as well as the rms boundary-layer profiles for velocity and temperature show a collapse onto a single curve during the whole compression stroke. This finding can serve as base for the development of novel wall heat transfer models. Finally, the DNS data showed that between 60% and 80% of the total wall heat losses during the compression stroke are attributed to convective transport due to wall-normal velocity fluctuations.

Interferometric study of heat transfer characteristics of Al2O3 and SiO2-based dilute nanofluids under simultaneously developing flow regime in compact channels

Performance evaluation of two types of nanofluids in influencing the heat transfer phenomena in the context of compact channels has been presented. Experiments have been conducted in forced convection regime for a range of Reynolds numbers. Three different types of coolant fluids i.e. de-ionized water (base fluid), Al2O3 and SiO2-based dilute nanofluids with volumetric concentrations of 0.005%, 0.01% and 0.02% have been employed. A Mach–Zehnder interferometer has been used for recording the real time projection data of the convective field. The interferometer has been operated in infinite as well as wedge fringe setting mode. The infinite fringe setting images have been employed for discussing the effect of nanoparticles on phenomena like thermal boundary layer profiles, changes in the thickness of thermal boundary layers, etc. The wedge interferograms have been employed for quantitative analysis. A direct comparison of heat transfer characteristics of Al2O3 and SiO2 nanoparticles has been presented on the basis of their relative influence on the thickness of thermal boundary layers, temperature gradients and the resultant heat transfer rates. The results of the study clearly reveal the effectiveness and higher heat transfer characteristics of Al2O3 nanoparticles than that of SiO2. It is seen that the Al2O3 nanoparticles have greater ability in disrupting the thermal boundary layer profiles and lead to higher percentage reduction in the thickness of thermal boundary layers. The resultant heat transfer coefficients with Al2O3-based dilute nanofluids have been found to be significantly higher than the base fluid and the SiO2-based nanofluids for any given volumetric concentration. It is experimentally demonstrated that the phenomena like increased thermal conductivity, boundary layer disruptions and advection effects primarily control the heat transfer rates in the lower range of Reynolds numbers (Re<500) while at higher Reynolds numbers, it is mainly the advection effects that influence the heat transfer coefficients in the range of volume concentrations of dilute nanofluids employed in the experiments.

Validation of the k–ω turbulence model for the thermal boundary layer profile of effusive cooled walls

The choice of materials for rocket chamber walls is limited by its thermal resistance. The thermal loads can be reduced substantially by the blowing out of gases through a porous surface. The k–ω-based turbulence models for computational fluid dynamic simulations are designed for smooth, non-permeable walls and have to be adjusted to account for the influence of injected fluids. Wilcox proposed therefore an extension for the k–ω turbulence model for the correct prediction of turbulent boundary layer velocity profiles. In this study, this extension is validated against experimental thermal boundary layer data from the Thermosciences Division of the Department of Mechanical Engineering from the Stanford University. All simulations are performed with a finite volume-based in-house code of the German Aerospace Center. Several simulations with different blowing settings were conducted and discussed in comparison to the results of the original model and in comparison to an additional roughness implementation. This study has permitted to understand that velocity profile corrections are necessary in contrast to additional roughness corrections to predict the correct thermal boundary layer profile of effusive cooled walls. Finally, this approach is applied to a two-dimensional simulation of an effusive cooled rocket chamber wall.

Interferometry-based whole field investigation of heat transfer characteristics of dilute nanofluids

The present work is concerned with understanding the heat transfer characteristics of dilute nanofluids using laser-based non-intrusive measurement techniques. Experiments have been conducted on a vertical heated flat plate immersed in nanofluids of varying concentrations, with water as the base fluid. Natural convection regime of the heat transfer between the flat plate and the ambient fluid medium is considered. Aluminium oxide (Al2O3) nanoparticles with volume concentrations in the range 0.005–0.02% are dispersed in the basefluid to prepare nanofluids of dilute concentrations. The phenomenon of convective heat transfer between the flat plate and the ambient fluid medium has been mapped using laser interferometry. Real time evolution of thermal boundary layers as a function of concentration of nanofluids has been recorded in order to understand the plausible role of nano particles in the heat transfer enhancement. Recorded interferograms have then been quantitatively analyzed to retrieve the two-dimensional temperature fields in each case and local distributions of heat transfer coefficients have been estimated. Results have been presented in the form of interferometric images of convective field in the vicinity of the flat plate, contours of whole field temperature distribution and local variation of heat transfer coefficient as a function of nanofluid concentration. The thermal boundary layer profiles as recorded using interferometry clearly reveal the effects of varying nanofluid concentration. In quantitative terms, an enhancement of about 21% in the heat transfer coefficient was observed for 0.02% concentration of nanofluid as compared to the base fluid whereas the Nusselt number was seen to deteriorate with increasing concentration of nanofluids.

The Time-Dependent Similarity Solutions of Boundary Layer Equations of Power-Law Fluids with Non-Isothermal Surface

Unsteady, two dimensional boundary layer flows over a heated surface of power-law fluids are investigated. Surface temperature is assumed to have o power-law variation with the time and the distance. Similarity transformation is applied to the partial differential equation system with three independent variables is reduced into an ordinary differential equations systems. Numerical solutions of non-linear differential equations are found by using a finite difference scheme. Solutions are obtained for boundary layer flow velocity and thermal boundary layer profile. Effects of flow behavior index, Prandtl number, suction-injection parameter and surface temperature exponent with the time and the distance are outlined in the figures.

Comparison between two- and three-dimensional Rayleigh–Bénard convection

AbstractTwo-dimensional and three-dimensional Rayleigh–Bénard convection is compared using results from direct numerical simulations and previous experiments. The phase diagrams for both cases are reviewed. The differences and similarities between two- and three-dimensional convection are studied using $Nu(Ra)$ for $\mathit{Pr}= 4. 38$ and $\mathit{Pr}= 0. 7$ and $Nu(Pr)$ for $Ra$ up to $1{0}^{8} $. In the $Nu(Ra)$ scaling at higher $Pr$, two- and three-dimensional convection is very similar, differing only by a constant factor up to $\mathit{Ra}= 1{0}^{10} $. In contrast, the difference is large at lower $Pr$, due to the strong roll state dependence of $Nu$ in two dimensions. The behaviour of $Nu(Pr)$ is similar in two and three dimensions at large $Pr$. However, it differs significantly around $\mathit{Pr}= 1$. The Reynolds number values are consistently higher in two dimensions and additionally converge at large $Pr$. Finally, the thermal boundary layer profiles are compared in two and three dimensions.