Three dimensional convection phenomenon Notes

: Three dimensional convection phenomenon Notes

 Page 1


Objectives_template
file:///G|/optical_measurement/lecture22/22_1.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
 
The Lecture Contains:
Convection in a Horizontal Differentially Heated Fluid Layer
Overview
Motivation
Rayleigh-Benard Convection
Experimental Details
Uncertainity and Measurement Errors
Image Processing and Data Reduction
Three-Dimensional Reconstruction Algorithm
Results and Discussion
Closure
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


Objectives_template
file:///G|/optical_measurement/lecture22/22_1.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
 
The Lecture Contains:
Convection in a Horizontal Differentially Heated Fluid Layer
Overview
Motivation
Rayleigh-Benard Convection
Experimental Details
Uncertainity and Measurement Errors
Image Processing and Data Reduction
Three-Dimensional Reconstruction Algorithm
Results and Discussion
Closure
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_2.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
 
Convection in a Horizontal Differentially Heated Fluid Layer 
Overview
An experimental study of Rayleigh-Benard convection in an intermediate aspect ratio box that is
square in plan is reported. An intermediate range of Rayleigh numbers has been considered in the
study. The fluid employed is air. A Mach-Zehnder interferometer is used to collect the line-of-sight
projections of the temperature field in the form of interferometric fringes. Images have been recorded
after a sufficient time has elapsed for the initial transients to have been eliminated. Interferograms
have been collected from four to six view angles. These are used to obtain the three-dimensional
temperature field inside the cavity by using tomography. The MART algorithm has been used for the
inversion of the projection data. The convergence of the iterative inversion procedure was
unambiguous and asymptotic. The reconstructed temperature field with a subset of the total data was
found to be consistent with the remaining unused projections.
Result for two Rayleigh numbers, namely 1.39 x 10
4
 and 4.02 x 10
4
 have been reported. These were
found to correspond to two distinct flow regimes. At these Rayleigh numbers, a well-defined steady
state was not observed. At the lower Rayleigh number, the fringes away from the wall showed mild
unsteadiness. At the higher Rayleigh number, the fringes were found to switch between two patterns.
Result for the dominate mode alone have been presented for this problem. At a Rayleigh number of
1.39x10
4
 , three-dimensional flow structures, whose influence is equivalent to longitudinal rolls have
been observed. At a Rayleigh number of 4.02 x 10
4
, cubic cells have been noted in the cavity. The
associated flow pattern is inferred to be a plume rising from the heated plate. The local Nusselt
number variation is seen to be consistent with the observed flow patterns.
Motivation
Rayleigh-Benard convection in horizontal fluid layers is a problem of fundamental as well as practical
importance. The flow pattern associated with this configuration shows a sequence of transitions from
steady laminar to unsteady flow and ultimately to turbulence. This configuration has been studied by
analytical and computational techniques as well as by experiments to understand the physics involved
in the transition phenomena. Although extensive work has been reported, many questions remain to
be answered. Many of the global features observed by numerical solutions are supported by
experimental observations. However, a closer comparison in terms of thermal field and convection
patterns remains to be carried out. With renewed interest in understanding nonlinear systems and
simultaneously the availability of powerful computers, there has been a revival of interest in Rayleigh-
Benard convection. The experimental technique has also been strengthened by the availability of
optical methods to visualize the flow phenomena and computers for data storage, processing, and
analysis.
Interferometric study of Rayleigh-Benard convection for two Rayleigh numbers (1.39x10
4
 and 4.02 x
10
4
) is reported in the present work. The cavity is square in plan and aspect ratio employed leads to
an intermediate aspect ratio box. The aspect ratio is defined as the ratio of the horizontal dimension
to the height of the cavity. The fluid considered is air. Results have been presented for flow patterns
that develop at long-time that are after the initial transients have been eliminated. Interferograms
collected from several line-of-sight projections have been processed to reconstruct the complete
three-dimensional temperature field. The multiplicative algebraic reconstruction technique in a
modified form called AVMART has been used as the preferred tomographic algorithm (AVMART)
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


Objectives_template
file:///G|/optical_measurement/lecture22/22_1.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
 
The Lecture Contains:
Convection in a Horizontal Differentially Heated Fluid Layer
Overview
Motivation
Rayleigh-Benard Convection
Experimental Details
Uncertainity and Measurement Errors
Image Processing and Data Reduction
Three-Dimensional Reconstruction Algorithm
Results and Discussion
Closure
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_2.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
 
Convection in a Horizontal Differentially Heated Fluid Layer 
Overview
An experimental study of Rayleigh-Benard convection in an intermediate aspect ratio box that is
square in plan is reported. An intermediate range of Rayleigh numbers has been considered in the
study. The fluid employed is air. A Mach-Zehnder interferometer is used to collect the line-of-sight
projections of the temperature field in the form of interferometric fringes. Images have been recorded
after a sufficient time has elapsed for the initial transients to have been eliminated. Interferograms
have been collected from four to six view angles. These are used to obtain the three-dimensional
temperature field inside the cavity by using tomography. The MART algorithm has been used for the
inversion of the projection data. The convergence of the iterative inversion procedure was
unambiguous and asymptotic. The reconstructed temperature field with a subset of the total data was
found to be consistent with the remaining unused projections.
Result for two Rayleigh numbers, namely 1.39 x 10
4
 and 4.02 x 10
4
 have been reported. These were
found to correspond to two distinct flow regimes. At these Rayleigh numbers, a well-defined steady
state was not observed. At the lower Rayleigh number, the fringes away from the wall showed mild
unsteadiness. At the higher Rayleigh number, the fringes were found to switch between two patterns.
Result for the dominate mode alone have been presented for this problem. At a Rayleigh number of
1.39x10
4
 , three-dimensional flow structures, whose influence is equivalent to longitudinal rolls have
been observed. At a Rayleigh number of 4.02 x 10
4
, cubic cells have been noted in the cavity. The
associated flow pattern is inferred to be a plume rising from the heated plate. The local Nusselt
number variation is seen to be consistent with the observed flow patterns.
Motivation
Rayleigh-Benard convection in horizontal fluid layers is a problem of fundamental as well as practical
importance. The flow pattern associated with this configuration shows a sequence of transitions from
steady laminar to unsteady flow and ultimately to turbulence. This configuration has been studied by
analytical and computational techniques as well as by experiments to understand the physics involved
in the transition phenomena. Although extensive work has been reported, many questions remain to
be answered. Many of the global features observed by numerical solutions are supported by
experimental observations. However, a closer comparison in terms of thermal field and convection
patterns remains to be carried out. With renewed interest in understanding nonlinear systems and
simultaneously the availability of powerful computers, there has been a revival of interest in Rayleigh-
Benard convection. The experimental technique has also been strengthened by the availability of
optical methods to visualize the flow phenomena and computers for data storage, processing, and
analysis.
Interferometric study of Rayleigh-Benard convection for two Rayleigh numbers (1.39x10
4
 and 4.02 x
10
4
) is reported in the present work. The cavity is square in plan and aspect ratio employed leads to
an intermediate aspect ratio box. The aspect ratio is defined as the ratio of the horizontal dimension
to the height of the cavity. The fluid considered is air. Results have been presented for flow patterns
that develop at long-time that are after the initial transients have been eliminated. Interferograms
collected from several line-of-sight projections have been processed to reconstruct the complete
three-dimensional temperature field. The multiplicative algebraic reconstruction technique in a
modified form called AVMART has been used as the preferred tomographic algorithm (AVMART)
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_2.htm[5/7/2012 12:31:24 PM]
 
 
Page 4


Objectives_template
file:///G|/optical_measurement/lecture22/22_1.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
 
The Lecture Contains:
Convection in a Horizontal Differentially Heated Fluid Layer
Overview
Motivation
Rayleigh-Benard Convection
Experimental Details
Uncertainity and Measurement Errors
Image Processing and Data Reduction
Three-Dimensional Reconstruction Algorithm
Results and Discussion
Closure
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_2.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
 
Convection in a Horizontal Differentially Heated Fluid Layer 
Overview
An experimental study of Rayleigh-Benard convection in an intermediate aspect ratio box that is
square in plan is reported. An intermediate range of Rayleigh numbers has been considered in the
study. The fluid employed is air. A Mach-Zehnder interferometer is used to collect the line-of-sight
projections of the temperature field in the form of interferometric fringes. Images have been recorded
after a sufficient time has elapsed for the initial transients to have been eliminated. Interferograms
have been collected from four to six view angles. These are used to obtain the three-dimensional
temperature field inside the cavity by using tomography. The MART algorithm has been used for the
inversion of the projection data. The convergence of the iterative inversion procedure was
unambiguous and asymptotic. The reconstructed temperature field with a subset of the total data was
found to be consistent with the remaining unused projections.
Result for two Rayleigh numbers, namely 1.39 x 10
4
 and 4.02 x 10
4
 have been reported. These were
found to correspond to two distinct flow regimes. At these Rayleigh numbers, a well-defined steady
state was not observed. At the lower Rayleigh number, the fringes away from the wall showed mild
unsteadiness. At the higher Rayleigh number, the fringes were found to switch between two patterns.
Result for the dominate mode alone have been presented for this problem. At a Rayleigh number of
1.39x10
4
 , three-dimensional flow structures, whose influence is equivalent to longitudinal rolls have
been observed. At a Rayleigh number of 4.02 x 10
4
, cubic cells have been noted in the cavity. The
associated flow pattern is inferred to be a plume rising from the heated plate. The local Nusselt
number variation is seen to be consistent with the observed flow patterns.
Motivation
Rayleigh-Benard convection in horizontal fluid layers is a problem of fundamental as well as practical
importance. The flow pattern associated with this configuration shows a sequence of transitions from
steady laminar to unsteady flow and ultimately to turbulence. This configuration has been studied by
analytical and computational techniques as well as by experiments to understand the physics involved
in the transition phenomena. Although extensive work has been reported, many questions remain to
be answered. Many of the global features observed by numerical solutions are supported by
experimental observations. However, a closer comparison in terms of thermal field and convection
patterns remains to be carried out. With renewed interest in understanding nonlinear systems and
simultaneously the availability of powerful computers, there has been a revival of interest in Rayleigh-
Benard convection. The experimental technique has also been strengthened by the availability of
optical methods to visualize the flow phenomena and computers for data storage, processing, and
analysis.
Interferometric study of Rayleigh-Benard convection for two Rayleigh numbers (1.39x10
4
 and 4.02 x
10
4
) is reported in the present work. The cavity is square in plan and aspect ratio employed leads to
an intermediate aspect ratio box. The aspect ratio is defined as the ratio of the horizontal dimension
to the height of the cavity. The fluid considered is air. Results have been presented for flow patterns
that develop at long-time that are after the initial transients have been eliminated. Interferograms
collected from several line-of-sight projections have been processed to reconstruct the complete
three-dimensional temperature field. The multiplicative algebraic reconstruction technique in a
modified form called AVMART has been used as the preferred tomographic algorithm (AVMART)
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_2.htm[5/7/2012 12:31:24 PM]
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_3.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
  
Rayleigh-Benard Convection
The present state-of-understanding of Rayleigh-Benard convection is discussed below. In the simplest
form, the flow configuration is comprised of a horizontal fluid layer confined between a pair of parallel
horizontal plates. The fluid is differentially heated by maintaining the lower surface at a higher
temperature compared to the top. This situation produces a top-heavy arrangement that is unstable.
The dimensionless quantity that characterized the buoyancy-driven flow is the Rayleigh number
defined as
(17)
When Ra is below a critical value, the gravitational potential is not sufficient to overcome the viscous
forces within the fluid layer. For Rayleigh numbers above the critical value, a steady flow is
established. Subsequently, flow undergoes a sequence of transitions, finally resulting in turbulence.
Transitions in Rayleigh-Benard convection depend on a Rayleigh number, a Prandtl number, and the
cavity aspect ratio. Additionally, there is an effect of the geometric structure of the side walls being
straight or curved [113]. The present discussion is restricted to a rectangular cavity. For a fluid layer
with an infinite aspect ratio, the first transition, namely the onset of fluid motion, occurs at a Rayleigh
number of 1708, irrespective of the Prandtl number. The associated flow pattern is in the form of
hexagonal cells. The general effect of lowering the aspect ratio is to stabilize the flow due to the
presence of the side walls and thus increase the critical Rayleigh number.  All subsequent transitions
are Prandtl number dependent.  The present discussion is devoted to Prandtl numbers in the range
0.7-7, for which some general conclusions can be drawn.
Flow patterns in rectangular cavities can be divided into three main categories, depending on the
aspect ratio.  These are small  intermediate ,  and large 
aspect ratio boxes.  Transition and chaos in a small aspect ratio enclosure with water has been
experimentally studied by Nasuno et al. Their data is in good agreement with the stability diagram of
Busse and Clever. In a large aspect ratio enclosure, it has been shown that flow beyond the critical
Rayleigh number is always time-dependent and non-periodic (see Ahlers and Behringer). In contrast,
a large number of bifurcations have been recorded both experimentally as well as in numerical
studies in small aspect ratio enclosures.  Information regarding intermediate aspect ratio enclosures is
sparse. The transition sequence appears to be via the formation of longitudinal rolls that are aligned
with the shorter side, polygonal cells; roll-loss and displacement; and finally towards turbulence.
When the Rayleigh number is close to the critical value for the onset of convection, hexagonal
convection cells have been observed both experimentally and in computation [113]. With a further
increase in the Rayleigh number, stable two-dimensional longitudinal rolls have been observed. 
Krishnamurti [120] is one of the earliest authors to conduct an experimental study and observe roll
patterns.  Much later, Kessler [121] obtained steady rolls formation through a numerical simulation. 
With further increase in the Rayleigh number, the two-dimensional rolls were seen to bifurcate slowly
to three-dimensional rolls, showing variation in shape along the roll axis.  The three-dimensional rolls
were found to be steady over a range of Rayleigh numbers.  For further increase in the Rayleigh
number, a loss-of-roll phenomena was observed. Kirchartz and Oertel [116] have shown for a box of
small aspect ratio the transition from four rolls to three rolls and finally to two rolls.  Simultaneously,
three-dimensional rolls become unstable and a periodic motion of the roll system begins along its
axis.  The critical Rayleigh number for the onset of oscillatory rolls is shown to be in the range of a
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 5


Objectives_template
file:///G|/optical_measurement/lecture22/22_1.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
 
The Lecture Contains:
Convection in a Horizontal Differentially Heated Fluid Layer
Overview
Motivation
Rayleigh-Benard Convection
Experimental Details
Uncertainity and Measurement Errors
Image Processing and Data Reduction
Three-Dimensional Reconstruction Algorithm
Results and Discussion
Closure
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_2.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
 
Convection in a Horizontal Differentially Heated Fluid Layer 
Overview
An experimental study of Rayleigh-Benard convection in an intermediate aspect ratio box that is
square in plan is reported. An intermediate range of Rayleigh numbers has been considered in the
study. The fluid employed is air. A Mach-Zehnder interferometer is used to collect the line-of-sight
projections of the temperature field in the form of interferometric fringes. Images have been recorded
after a sufficient time has elapsed for the initial transients to have been eliminated. Interferograms
have been collected from four to six view angles. These are used to obtain the three-dimensional
temperature field inside the cavity by using tomography. The MART algorithm has been used for the
inversion of the projection data. The convergence of the iterative inversion procedure was
unambiguous and asymptotic. The reconstructed temperature field with a subset of the total data was
found to be consistent with the remaining unused projections.
Result for two Rayleigh numbers, namely 1.39 x 10
4
 and 4.02 x 10
4
 have been reported. These were
found to correspond to two distinct flow regimes. At these Rayleigh numbers, a well-defined steady
state was not observed. At the lower Rayleigh number, the fringes away from the wall showed mild
unsteadiness. At the higher Rayleigh number, the fringes were found to switch between two patterns.
Result for the dominate mode alone have been presented for this problem. At a Rayleigh number of
1.39x10
4
 , three-dimensional flow structures, whose influence is equivalent to longitudinal rolls have
been observed. At a Rayleigh number of 4.02 x 10
4
, cubic cells have been noted in the cavity. The
associated flow pattern is inferred to be a plume rising from the heated plate. The local Nusselt
number variation is seen to be consistent with the observed flow patterns.
Motivation
Rayleigh-Benard convection in horizontal fluid layers is a problem of fundamental as well as practical
importance. The flow pattern associated with this configuration shows a sequence of transitions from
steady laminar to unsteady flow and ultimately to turbulence. This configuration has been studied by
analytical and computational techniques as well as by experiments to understand the physics involved
in the transition phenomena. Although extensive work has been reported, many questions remain to
be answered. Many of the global features observed by numerical solutions are supported by
experimental observations. However, a closer comparison in terms of thermal field and convection
patterns remains to be carried out. With renewed interest in understanding nonlinear systems and
simultaneously the availability of powerful computers, there has been a revival of interest in Rayleigh-
Benard convection. The experimental technique has also been strengthened by the availability of
optical methods to visualize the flow phenomena and computers for data storage, processing, and
analysis.
Interferometric study of Rayleigh-Benard convection for two Rayleigh numbers (1.39x10
4
 and 4.02 x
10
4
) is reported in the present work. The cavity is square in plan and aspect ratio employed leads to
an intermediate aspect ratio box. The aspect ratio is defined as the ratio of the horizontal dimension
to the height of the cavity. The fluid considered is air. Results have been presented for flow patterns
that develop at long-time that are after the initial transients have been eliminated. Interferograms
collected from several line-of-sight projections have been processed to reconstruct the complete
three-dimensional temperature field. The multiplicative algebraic reconstruction technique in a
modified form called AVMART has been used as the preferred tomographic algorithm (AVMART)
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_2.htm[5/7/2012 12:31:24 PM]
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_3.htm[5/7/2012 12:31:24 PM]
 Module 4: Interferometry
 Lecture 22: Three dimensional convection phenomenon
  
Rayleigh-Benard Convection
The present state-of-understanding of Rayleigh-Benard convection is discussed below. In the simplest
form, the flow configuration is comprised of a horizontal fluid layer confined between a pair of parallel
horizontal plates. The fluid is differentially heated by maintaining the lower surface at a higher
temperature compared to the top. This situation produces a top-heavy arrangement that is unstable.
The dimensionless quantity that characterized the buoyancy-driven flow is the Rayleigh number
defined as
(17)
When Ra is below a critical value, the gravitational potential is not sufficient to overcome the viscous
forces within the fluid layer. For Rayleigh numbers above the critical value, a steady flow is
established. Subsequently, flow undergoes a sequence of transitions, finally resulting in turbulence.
Transitions in Rayleigh-Benard convection depend on a Rayleigh number, a Prandtl number, and the
cavity aspect ratio. Additionally, there is an effect of the geometric structure of the side walls being
straight or curved [113]. The present discussion is restricted to a rectangular cavity. For a fluid layer
with an infinite aspect ratio, the first transition, namely the onset of fluid motion, occurs at a Rayleigh
number of 1708, irrespective of the Prandtl number. The associated flow pattern is in the form of
hexagonal cells. The general effect of lowering the aspect ratio is to stabilize the flow due to the
presence of the side walls and thus increase the critical Rayleigh number.  All subsequent transitions
are Prandtl number dependent.  The present discussion is devoted to Prandtl numbers in the range
0.7-7, for which some general conclusions can be drawn.
Flow patterns in rectangular cavities can be divided into three main categories, depending on the
aspect ratio.  These are small  intermediate ,  and large 
aspect ratio boxes.  Transition and chaos in a small aspect ratio enclosure with water has been
experimentally studied by Nasuno et al. Their data is in good agreement with the stability diagram of
Busse and Clever. In a large aspect ratio enclosure, it has been shown that flow beyond the critical
Rayleigh number is always time-dependent and non-periodic (see Ahlers and Behringer). In contrast,
a large number of bifurcations have been recorded both experimentally as well as in numerical
studies in small aspect ratio enclosures.  Information regarding intermediate aspect ratio enclosures is
sparse. The transition sequence appears to be via the formation of longitudinal rolls that are aligned
with the shorter side, polygonal cells; roll-loss and displacement; and finally towards turbulence.
When the Rayleigh number is close to the critical value for the onset of convection, hexagonal
convection cells have been observed both experimentally and in computation [113]. With a further
increase in the Rayleigh number, stable two-dimensional longitudinal rolls have been observed. 
Krishnamurti [120] is one of the earliest authors to conduct an experimental study and observe roll
patterns.  Much later, Kessler [121] obtained steady rolls formation through a numerical simulation. 
With further increase in the Rayleigh number, the two-dimensional rolls were seen to bifurcate slowly
to three-dimensional rolls, showing variation in shape along the roll axis.  The three-dimensional rolls
were found to be steady over a range of Rayleigh numbers.  For further increase in the Rayleigh
number, a loss-of-roll phenomena was observed. Kirchartz and Oertel [116] have shown for a box of
small aspect ratio the transition from four rolls to three rolls and finally to two rolls.  Simultaneously,
three-dimensional rolls become unstable and a periodic motion of the roll system begins along its
axis.  The critical Rayleigh number for the onset of oscillatory rolls is shown to be in the range of a
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture22/22_3.htm[5/7/2012 12:31:24 PM]
Rayleigh number of 30000 for air (Pr = 0.71) [121].  This critical Rayleigh number increases with the
increase of the Prandtl number.  The frequency of oscillation is not a strong function of the Rayleigh
number, but increases slowly with increase in the Rayleigh number. Kessler [121] has also observed
the exchange of mass between different rolls.  This is due to a periodic motion in the location of the
vertically upward and downward flow.
 
 
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