scholarly journals Solution of the Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Georgios Karamanis ◽  
Marc Hodes ◽  
Toby Kirk ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Boundary Condition ◽  
Temperature Profile ◽  
Mixed Boundary ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
The Other ◽  
Inlet Temperature ◽  
Textured Surface ◽  
Parallel Plates ◽  
Time Required

We consider convective heat transfer for laminar flow of liquid between parallel plates that are textured with isothermal ridges oriented parallel to the flow. Three different flow configurations are analyzed: one plate textured and the other one smooth; both plates textured and the ridges aligned; and both plates textured, but the ridges staggered by half a pitch. The liquid is assumed to be in the Cassie state on the textured surface(s), to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. Heat is exchanged with the liquid either through the ridges of one plate with the other plate adiabatic, or through the ridges of both plates. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). Axial conduction is neglected and the inlet temperature profile is arbitrary. We solve for the three-dimensional developing temperature profile assuming a hydrodynamically developed flow, i.e., we consider the Graetz–Nusselt problem. Using the method of separation of variables, the thermal problem is essentially reduced to a two-dimensional eigenvalue problem in the transverse coordinates, which is solved numerically. Expressions for the local Nusselt number and those averaged over the period of the ridges in the developing and fully developed regions are provided. Nusselt numbers averaged over the period and length of the domain are also provided. Our approach enables the aforementioned quantities to be computed in a small fraction of the time required by a general computational fluid dynamics (CFD) solver.

Nusselt Numbers for Fully-Developed Flow Between Parallel Plates With One Plate Textured With Isothermal Parallel Ridges

2016 ◽  
Author(s):  
Georgios Karamanis ◽  
Marc Hodes ◽  
Toby Kirk ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Boundary Condition ◽  
Nusselt Number ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Liouville Problem ◽  
Textured Surface ◽  
First Eigenvalue ◽  
Parallel Plates ◽  
Fully Developed Flow ◽  
Streamwise Location

We develop a semi-analytical solution for the Nusselt number for fully-developed flow of liquid between parallel plates, one of which is textured with isothermal parallel ridges. The opposite plate is smooth and adiabatic. The liquid is assumed to be in the Cassie state on the textured surface, on which a mixed boundary condition of no slip on the ridges and no shear along menisci applies. An existing solution for the velocity field is valid. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface. Given the nature of the isothermal boundary condition, the analysis concerns a three-dimensional developing temperature profile, and the results are obtained for a streamwise location that tends to infinity. We assume that the temperature field is governed by an infinite sum of the product of a function of the streamwise coordinate and a second function of the spanwise co-ordinates. The latter functions are eigenfunctions satisfying a two-dimensional Sturm-Liouville problem from which the eigenvalues follow. The fully-developed Nusselt number follows from the first eigenvalue.

scholarly journals Solution of the Extended Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
Georgios Karamanis ◽  
Marc Hodes ◽  
Toby Kirk ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Boundary Condition ◽  
Viscous Dissipation ◽  
Heat Generation ◽  
Mixed Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Separation Of Variables ◽  
Infinite Length ◽  
Textured Surface ◽  
Parallel Plates ◽  
Volumetric Heat Generation

We consider convective heat transfer for laminar flow of liquid between parallel plates. The configurations analyzed are both plates textured with symmetrically aligned isothermal ridges oriented parallel to the flow, and one plate textured as such and the other one smooth and adiabatic. The liquid is assumed to be in the Cassie state on the textured surface(s) to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). We solve for the developing three-dimensional temperature profile resulting from a step change of the ridge temperature in the streamwise direction assuming a hydrodynamically developed flow. Axial conduction is accounted for, i.e., we consider the extended Graetz–Nusselt problem; therefore, the domain is of infinite length. The effects of viscous dissipation and (uniform) volumetric heat generation are also captured. Using the method of separation of variables, the homogeneous part of the thermal problem is reduced to a nonlinear eigenvalue problem in the transverse coordinates which is solved numerically. Expressions derived for the local and the fully developed Nusselt number along the ridge and that averaged over the composite interface in terms of the eigenvalues, eigenfunctions, Brinkman number, and dimensionless volumetric heat generation rate. Estimates are provided for the streamwise location where viscous dissipation effects become important.

Temperature Predictions and Comparison With Measurements for the Blade Leading Edge and Platform of a 1-1/2 Stage Transonic HP Turbine

2010 ◽  
Author(s):  
R. M. Mathison ◽  
M. B. Wishart ◽  
C. W. Haldeman ◽  
M. G. Dunn
Keyword(s):  
Boundary Condition ◽  
Temperature Profile ◽  
Short Duration ◽  
Computational Models ◽  
Current Model ◽  
Leading Edge ◽  
Inlet Temperature ◽  
Fluid Temperature ◽  
One Dimensional ◽  
Blade Leading Edge

A series of computational predictions generated using FINE/Turbo are compared with data to investigate implementation techniques available for predicting temperature migration through a turbine stage. The experimental results used for comparison are from a one-and-one-half stage turbine operating at design-corrected conditions in a short-duration facility. Measurements of the boundary conditions are used to set up the computational models, and the predicted temperatures are compared to measured fluid temperatures at the blade leading edge and just above the blade platform. Fluid temperature measurements have not previously been available for these locations in a transonic turbine operating at design-corrected conditions, so this represents a novel comparison. Accurate predictions for this short-duration turbine experiment require use of the iso-thermal wall boundary condition instead of an adiabatic boundary condition and accurate specification of the inlet temperature profile all the way to the wall. Predictions using the harmonic method agree with the temperatures measured for the blade leading edge from 65% to 95% span to within 1% normalized temperature data. Agreement over much of the rest of the leading edge is within 5% of the measured value. Comparisons at 5–10% span and for the blade platform show larger differences up to 10%, which indicates that the flow in this region is not fully captured by the prediction. This is not surprising since the purge cavity and platform leading edge features present in the experiment are treated as a smooth hub wall in the current simulation. This work represents a step towards the larger goal of accurately predicting surface heat-flux for the complicated environment of an operational engine as it is reproduced in a laboratory setting. The experiment upon which these computations are based includes realistic complications such as one-dimensional and two-dimensional inlet temperature profiles, a heavily film-cooled vane, and purge cooling. While the ultimate goal is to accurately handle all of these features, the current model focuses on the treatment of a subset of experiments performed for a one-dimensional radial inlet temperature profile and no cooling.

Three-Dimensional Flow in an Axial Turbine: Part 2—Profile Attenuation

1992 ◽  
Vol 114 (1) ◽  
pp. 71-78 ◽  
Author(s):  
D. Joslyn ◽  
R. Dring
Keyword(s):  
Temperature Profile ◽  
Three Dimensional ◽  
Inlet Temperature ◽  
Three Dimensional Flow ◽  
Turbine Inlet Temperature ◽  
Axial Turbine ◽  
Physical Mechanisms ◽  
Dimensional Flow ◽  
Surface Measurements ◽  
The Impact

This paper presents an exhaustive experimental documentation of the three-dimensional nature of the flow in a one-and-one-half stage axial turbine. The intent was to examine the flow within, and downstream of, both the stator and rotor airfoil rows so as to delineate the dominant physical mechanisms. Part 1 of this paper presented the aerodynamic results. Part 2 presents documentation of the mixing, or attenuation, of a simulated spanwise inlet temperature profile as it passed through the turbine, including: (1) the simulated combustor exit-turbine inlet temperature profile, (2) surface measurements on the airfoils and endwalls of the three airfoil rows, and (3) radial–circumferential distributions downstream of each airfoil. Although all three rows contributed to profile attenuation, the impact of the rotor was strongest.

An Exact Periodic Solution of a Hydromagnetic Flow of an Oldroyd Fluid in a Channel

1999 ◽  
Vol 66 (4) ◽  
pp. 974-977 ◽  
Author(s):  
R. N. Ray ◽  
A. Samad ◽  
T. K. Chaudhury
Keyword(s):  
Periodic Solution ◽  
Unsteady Flow ◽  
Velocity Gradient ◽  
Separation Of Variables ◽  
The Other ◽  
Parallel Plates ◽  
Hydromagnetic Flow ◽  
Mean Velocity ◽  
Oldroyd Fluid ◽  
Viscoelastic Parameters

The unsteady flow of a conducting Oldroyd fluid between two parallel plates, one of which is at rest and the other oscillating in its own plane with a constant mean velocity, has been investigated. Using separation of variables an exact periodic solution for the problem is obtained. Effects of viscoelastic parameters on velocity gradient, skin friction amplitude, and phase angle of the flow are discussed with graphs.

A Trace Gas Technique to Study Mixing in a Turbine Stage

1988 ◽  
Vol 110 (1) ◽  
pp. 38-43 ◽  
Author(s):  
H. D. Joslyn ◽  
R. P. Dring
Keyword(s):  
Temperature Profile ◽  
Large Scale ◽  
Three Dimensional ◽  
Axial Flow ◽  
Surface Flow ◽  
Trace Gas ◽  
Inlet Temperature ◽  
Turbine Stage ◽  
Radial Temperature ◽  
Gas Concentration

An experimental technique to study mixing in a turbine stage is demonstrated. An axisymmetric, radial temperature profile at the inlet to the first stator of a large-scale, low-speed, single-stage, axial flow turbine model is simulated with a radial trace gas concentration distribution. Mixing or redistribution of the inlet profile by three-dimensional aerodynamic mechanisms (other than temperature-driven mechanisms) is determined from trace gas concentration measurements made in both the stationary and rotating frames of reference at various locations through the turbine. The trace gas concentration contours generated are consistent with flow pitch angle measurements made downstream of the first stator and with surface flow visualization on the rotor airfoil and the hub endwall. It is demonstrated that this trace gas technique is well suited to quantify many aspects of the redistribution and diffusion of an inlet temperature profile as it is convected through a turbine stage.

Three-Dimensional Flow in an Axial Turbine: Part 1 — Aerodynamic Mechanisms

1990 ◽  
Author(s):  
David Joslyn ◽  
Robert Dring
Keyword(s):  
Flow Visualization ◽  
Temperature Profile ◽  
Three Dimensional ◽  
Surface Flow ◽  
Inlet Temperature ◽  
Three Dimensional Flow ◽  
Axial Turbine ◽  
Physical Mechanisms ◽  
Pressure Distributions ◽  
Flow Part

This paper presents an exhaustive experimental documentation of the three–dimensional nature of the flow in a one–and–one–half stage axial turbine. The intent was to examine the flow within, and downstream of, both the stator and rotor airfoil rows so as to delineate the dominant physical mechanisms. Part 1 of this paper presents the aerodynamic results including: (1) airfoil and endwall surface flow visualization, (2) fullspan airfoil pressure distributions, and (3) radial–circumferential distributions of the total and static pressures, and of the yaw and pitch angles in the flow. Part 2 of the paper presents results describing the mixing, or attenuation, of a simulated spanwise inlet temperature profile as it passed through the turbine. Although the flow in each airfoil row possessed a degree of three–dimensionality, that in the rotor was the strongest.

Three-Dimensional Flow in an Axial Turbine: Part 2 — Profile Attenuation

1990 ◽  
Author(s):  
David Joslyn ◽  
Robert Dring
Keyword(s):  
Temperature Profile ◽  
Three Dimensional ◽  
Inlet Temperature ◽  
Three Dimensional Flow ◽  
Turbine Inlet Temperature ◽  
Axial Turbine ◽  
Physical Mechanisms ◽  
Dimensional Flow ◽  
Surface Measurements ◽  
The Impact

This paper presents an exhaustive experimental documentation of the three–dimensional nature of the flow in a one–and–one–half stage axial turbine. The intent was to examine the flow within, and downstream of, both the stator and rotor airfoil rows so as to delineate the dominant physical mechanisms. Part 1 of this paper presented the aerodynamic results. Part 2 presents documentation of the mixing, or attenuation, of a simulated spanwise inlet temperature profile as it passed through the turbine including: (1) the simulated combustor exit–turbine inlet temperature profile, (2) surface measurements on the airfoils and endwalls of the three airfoil rows, and (3) radial–circumferential distributions downstream of each airfoil. Although all three rows contributed to profile attenuation, the impact of the rotor was strongest.

Temperature Predictions and Comparison With Measurements for the Blade Leading Edge and Platform of a 1 1/2 Stage Transonic HP Turbine

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Randall M. Mathison ◽  
Mark B. Wishart ◽  
Charles W. Haldeman ◽  
Michael G. Dunn
Keyword(s):  
Boundary Condition ◽  
Temperature Profile ◽  
Short Duration ◽  
Computational Models ◽  
Current Model ◽  
Leading Edge ◽  
Inlet Temperature ◽  
Fluid Temperature ◽  
One Dimensional ◽  
Blade Leading Edge

A series of computational predictions generated using FINE/TURBO are compared with data to investigate implementation techniques available for predicting temperature migration through a turbine stage. The experimental results used for comparison are from a one-and-one-half stage turbine operating at design-corrected conditions in a short-duration facility. Measurements of the boundary conditions are used to set up the computational models, and the predicted temperatures are compared with measured fluid temperatures at the blade leading edge and just above the blade platform. Fluid temperature measurements have not previously been available for these locations in a transonic turbine operating at design-corrected conditions, so this represents a novel comparison. Accurate predictions for this short-duration turbine experiment require use of the isothermal wall boundary condition instead of an adiabatic boundary condition and accurate specification of the inlet temperature profile all the way to the wall. Predictions using the harmonic method agree with the temperatures measured for the blade leading edge from 65% to 95% span to within 1% normalized temperature data. Agreement over much of the rest of the leading edge is within 5% of the measured value. Comparisons at 5–10% span and for the blade platform show larger differences up to 10%, which indicates that the flow in this region is not fully captured by the prediction. This is not surprising since the purge cavity and platform leading-edge features present in the experiment are treated as a smooth hub wall in the current simulation. This work represents a step toward the larger goal of accurately predicting surface heat-flux for the complicated environment of an operational engine as it is reproduced in a laboratory setting. The experiment upon which these computations are based includes realistic complications such as one-dimensional and two-dimensional inlet temperature profiles, a heavily film-cooled vane, and purge cooling. While the ultimate goal is to accurately handle all of these features, the current model focuses on the treatment of a subset of experiments performed for a one-dimensional radial inlet temperature profile and no cooling.

Three-Dimensional Flow in an Axial Turbine: Part 1—Aerodynamic Mechanisms

1992 ◽  
Vol 114 (1) ◽  
pp. 61-70 ◽  
Author(s):  
D. Joslyn ◽  
R. Dring
Keyword(s):  
Flow Visualization ◽  
Temperature Profile ◽  
Three Dimensional ◽  
Surface Flow ◽  
Inlet Temperature ◽  
Three Dimensional Flow ◽  
Axial Turbine ◽  
Physical Mechanisms ◽  
Pressure Distributions ◽  
Flow Part

This paper presents an exhaustive experimental documentation of the three-dimensional nature of the flow in a one-and-one-half stage axial turbine. The intent was to examine the flow within, and downstream of, both the stator and rotor airflow rows so as to delineate the dominant physical mechanisms. Part 1 of this paper presents the aerodynamic results including: (1) airflow and endwall surface flow visualization, (2) full-span airfoil pressure distributions, and (3) radial-circumferential distributions of the total and static pressures, and of the yaw and pitch angles in the flow. Part 2 of the paper presents results describing the mixing, or attenuation, of a simulated spanwise inlet temperature profile as it passed through the turbine. Although the flow in each airfoil row possessed a degree of three-dimensionality, that in the rotor was the strongest.

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