Introduction to numerical methods - Revision history

Tsaad at 21:02, 24 May 2007

2007-05-24T21:02:34Z

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	In physical modeling, we seek a set of equations or mathematical relations that allow us to close the governing equations. In turbulence modeling for example, one is interested in devising new equations for the extra unknowns that resulted from the averaging process.		In physical modeling, we seek a set of equations or mathematical relations that allow us to close the governing equations. In turbulence modeling for example, one is interested in devising new equations for the extra unknowns that resulted from the averaging process.

-	On the other hand, the focus in numerics is to devise efficient, robust, and reliable algorithms for the solution of PDEs. PDEs are a combination of differential terms (rates of change) that describe a conservation principle. Without loss in generality, all physical processes can be described by PDEs. Now, the CFD process requires the discretization of the governing PDEs, i.e. the derivation of equivalent algebraic relations that should faithfully represent the original PDEs. This is done by transforming each differential term into an approximate algebraic relation (see [[~~Generic_scalar_transport_equation~~]]).	+	On the other hand, the focus in numerics is to devise efficient, robust, and reliable algorithms for the solution of PDEs. PDEs are a combination of differential terms (rates of change) that describe a conservation principle. Without loss in generality, all physical processes can be described by PDEs. Now, the CFD process requires the discretization of the governing PDEs, i.e. the derivation of equivalent algebraic relations that should faithfully represent the original PDEs. This is done by transforming each differential term into an approximate algebraic relation (see [[Generic scalar transport equation]]).

	Deriving an original numerical algorithm is not only a mathematical challenge. The investigator should also bear in mind the physics behind the term that is being discretized. For example, there are various discretization schemes for the convection term (upwind, QUICK, SOU etc...) because of the special behavior of the convection process. Similarly, the diffusion, convection, and source terms have very specialized treatments.		Deriving an original numerical algorithm is not only a mathematical challenge. The investigator should also bear in mind the physics behind the term that is being discretized. For example, there are various discretization schemes for the convection term (upwind, QUICK, SOU etc...) because of the special behavior of the convection process. Similarly, the diffusion, convection, and source terms have very specialized treatments.

Tsaad: cleanup

2006-11-02T17:52:16Z

cleanup

← Older revision		Revision as of 17:52, 2 November 2006
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	Numerical methods are at the heart of the CFD process. Researchers dedicate their attention to two fundamental aspects in CFD; i.e. physical modeling and numerics.		Numerical methods are at the heart of the CFD process. Researchers dedicate their attention to two fundamental aspects in CFD; i.e. physical modeling and numerics.

-	In physical modeling, we seek a set of equations or mathematical relations that allow us to close the ~~given set of~~ equations. In turbulence modeling for example, one is interested in devising new equations for the extra unknowns that ~~result~~ from the averaging process.	+	In physical modeling, we seek a set of equations or mathematical relations that allow us to close the governing equations. In turbulence modeling for example, one is interested in devising new equations for the extra unknowns that resulted from the averaging process.

-	On the other hand, the focus in numerics is to devise efficient, robust, and reliable algorithms for the solution of PDEs. PDEs are a combination of differential terms (rates of change) that describe a conservation principle. Without loss in generality, all physical processes can be described by PDEs. Now, the CFD process requires the discretization of the governing PDEs, i.e. the derivation of equivalent algebraic relations that should faithfully represent the original ~~PDE~~. This is done by transforming each differential term ~~in the PDE~~ into an approximate algebraic relation (see [[Generic_scalar_transport_equation]]).	+	On the other hand, the focus in numerics is to devise efficient, robust, and reliable algorithms for the solution of PDEs. PDEs are a combination of differential terms (rates of change) that describe a conservation principle. Without loss in generality, all physical processes can be described by PDEs. Now, the CFD process requires the discretization of the governing PDEs, i.e. the derivation of equivalent algebraic relations that should faithfully represent the original PDEs. This is done by transforming each differential term into an approximate algebraic relation (see [[Generic_scalar_transport_equation]]).

	Deriving an original numerical algorithm is not only a mathematical challenge. The investigator should also bear in mind the physics behind the term that is being discretized. For example, there are various discretization schemes for the convection term (upwind, QUICK, SOU etc...) because of the special behavior of the convection process. Similarly, the diffusion, convection, and source terms have very specialized treatments.		Deriving an original numerical algorithm is not only a mathematical challenge. The investigator should also bear in mind the physics behind the term that is being discretized. For example, there are various discretization schemes for the convection term (upwind, QUICK, SOU etc...) because of the special behavior of the convection process. Similarly, the diffusion, convection, and source terms have very specialized treatments.

Tsaad: Removed all previous text as it was not related to numerical methods

2006-11-02T17:50:46Z

Removed all previous text as it was not related to numerical methods

← Older revision		Revision as of 17:50, 2 November 2006
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-	~~<p>[[Fluid]]s could be regarded as substances those on application~~ of ~~external force deform or show very little resistance to deformation. This deformation in~~ the ~~cases of liquid and gases is substantial~~. Both obey common laws of motion and both for most practical computational purposes can be regarded as continuum. Forces can be considered as body forces, example: forces due to gravity or they can be considered as surface forces example: surface tension.</p>	+	Numerical methods are at the heart of the CFD process. Researchers dedicate their attention to two fundamental aspects in CFD; i.e. physical modeling and numerics.
-	~~<p>~~	+
-	~~Based on speed and nature of flow, flows could be categorized into: <br>~~	+
-	*[[Laminar flow]] <br>	+
-	*[[Turbulent flow]] <br>	+
-	~~Or it could be categorized into: <br>~~	+
-	*[[Subsonic flow]] <br>	+
-	*[[Supersonic flow]] <br>	+
-	~~Based upon~~ their ~~[[Mach number]]. This Mach number could be used~~ to determine whether the flow shall be considered as [[Incompressible flow ]] or [[Compressible flow]] for computational purposes. For the flows of Mach number smaller than 0.3, we can safely assume them as incompressible for all computational purposes. </p>	+
-	~~<p>~~	+
-	~~Based upon the degree of effect upon the viscosity on the flow, the flows could be classified as [[Viscous flow]] or [[Inviscid flow]]~~ in ~~nature~~. ~~Fluids obeying [[Newton’s law]] are called [[Newtonian fluid \| Newtonian fluids]]~~.	+
-	~~Further based upon the phases of flow, the flows can be considered as [[multiphase]] or single phase flows~~. ~~</p>~~	+

-	~~== Control Volume Approach ==~~	+	In physical modeling, we seek a set of equations or mathematical relations that allow us to close the given set of equations. In turbulence modeling for example, one is interested in devising new equations for the extra unknowns that result from the averaging process.
-	~~<p>~~	+
-	~~Conservation laws can be applied~~ to given ~~quantity~~ of ~~fluid~~. ~~However~~, it is ~~more convenient to apply them on a small spatial region, called Control volume. For this purposes~~ the ~~geometry under consideration is subdivided into smaller sub-regions. Together they are called mesh or grid. And~~ the ~~approach used is called Control volume approach for solving the flow problem. <br>~~	+
-	~~The figure 1.1 shows one such computational domain~~. ~~<br>~~	+

-	~~[[Image:Img_nm_intro_01~~.~~jpg]] <br>~~	+	On the other hand, the focus in numerics is to devise efficient, robust, and reliable algorithms for the solution of PDEs. PDEs are a combination of differential terms (rates of change) that describe a conservation principle. Without loss in generality, all physical processes can be described by PDEs. Now, the CFD process requires the discretization of the governing PDEs, i.e. the derivation of equivalent algebraic relations that should faithfully represent the original PDE. This is done by transforming each differential term in the PDE into an approximate algebraic relation (see [[Generic_scalar_transport_equation]]).
-	~~Figure 1~~.~~1 <br>~~	+
-	~~</p>~~	+
-	~~== Conservation Principles ==~~	+
-	~~== Conservation~~ of ~~scalar quantities ==~~	+
-	~~== Simple Flows ==~~	+
-	~~===~~ [[~~Incompressible flow~~]] ~~===~~	+
-	~~=== [[Inviscid flow \| Euler or inviscid flow]] ===~~	+
-	~~=== [[Potential flow]] ===~~	+
-	~~=== [[Stokes flow \| Stokes or creeping flow]] ===~~	+

-	~~----~~	+	Deriving an original numerical algorithm is not only a mathematical challenge. The investigator should also bear in mind the physics behind the term that is being discretized. For example, there are various discretization schemes for the convection term (upwind, QUICK, SOU etc...) because of the special behavior of the convection process. Similarly, the diffusion, convection, and source terms have very specialized treatments.
-	~~<i> Return to [[Numerical methods \| Numerical Methods]] </i>~~	+

Zahidmath at 07:05, 21 November 2005

2005-11-21T07:05:19Z

← Older revision		Revision as of 07:05, 21 November 2005
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-	<p>[[Fluid]]s could be regarded as substances those on application of external force deform or show very little resistance to deformation. This deformation in the cases of liquid and gases is substantial. Both obey common ~~lows~~ of motion and both for most practical computational purposes can be regarded as continuum. Forces can be considered as body forces, example: forces due to gravity or they can be considered as surface forces example: surface tension.</p>	+	<p>[[Fluid]]s could be regarded as substances those on application of external force deform or show very little resistance to deformation. This deformation in the cases of liquid and gases is substantial. Both obey common laws of motion and both for most practical computational purposes can be regarded as continuum. Forces can be considered as body forces, example: forces due to gravity or they can be considered as surface forces example: surface tension.</p>
	<p>		<p>
	Based on speed and nature of flow, flows could be categorized into: <br>		Based on speed and nature of flow, flows could be categorized into: <br>

Praveen at 09:45, 5 October 2005

2005-10-05T09:45:54Z

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	== Simple Flows ==		== Simple Flows ==
	=== [[Incompressible flow]] ===		=== [[Incompressible flow]] ===
-	=== [[Euler or inviscid flow]] ===	+	=== [[Inviscid flow \| Euler or inviscid flow]] ===
	=== [[Potential flow]] ===		=== [[Potential flow]] ===
-	=== [[Stokes or creeping flow]] ===	+	=== [[Stokes flow \| Stokes or creeping flow]] ===

	----		----
	<i> Return to [[Numerical methods \| Numerical Methods]] </i>		<i> Return to [[Numerical methods \| Numerical Methods]] </i>

Zxaar at 06:16, 3 October 2005

2005-10-03T06:16:51Z

← Older revision		Revision as of 06:16, 3 October 2005
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	=== [[Potential flow]] ===		=== [[Potential flow]] ===
	=== [[Stokes or creeping flow]] ===		=== [[Stokes or creeping flow]] ===
		+
		+	----
		+	<i> Return to [[Numerical methods \| Numerical Methods]] </i>

Zxaar at 00:09, 14 September 2005

2005-09-14T00:09:49Z

← Older revision		Revision as of 00:09, 14 September 2005
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	Based upon their [[Mach number]]. This Mach number could be used to determine whether the flow shall be considered as [[Incompressible flow ]] or [[Compressible flow]] for computational purposes. For the flows of Mach number smaller than 0.3, we can safely assume them as incompressible for all computational purposes. </p>		Based upon their [[Mach number]]. This Mach number could be used to determine whether the flow shall be considered as [[Incompressible flow ]] or [[Compressible flow]] for computational purposes. For the flows of Mach number smaller than 0.3, we can safely assume them as incompressible for all computational purposes. </p>
	<p>		<p>
-	Based upon the degree of effect upon the viscosity on the flow, the flows could be classified as [[Viscous flow]] or [[Inviscid flow]] in nature. Fluids obeying [[Newton’s law]] are called [[Newtonian]].	+	Based upon the degree of effect upon the viscosity on the flow, the flows could be classified as [[Viscous flow]] or [[Inviscid flow]] in nature. Fluids obeying [[Newton’s law]] are called [[Newtonian fluid \| Newtonian fluids]].
	Further based upon the phases of flow, the flows can be considered as [[multiphase]] or single phase flows. </p>		Further based upon the phases of flow, the flows can be considered as [[multiphase]] or single phase flows. </p>

Zxaar at 00:06, 14 September 2005

2005-09-14T00:06:50Z

← Older revision		Revision as of 00:06, 14 September 2005
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-	<p>~~Fluids~~ could be regarded as substances those on application of external force deform or show very little resistance to deformation. This deformation in the cases of liquid and gases is substantial. Both obey common lows of motion and both for most practical computational purposes can be regarded as continuum. Forces can be considered as body forces, example: forces due to gravity or they can be considered as surface forces example: surface tension.</p>	+	<p>[[Fluid]]s could be regarded as substances those on application of external force deform or show very little resistance to deformation. This deformation in the cases of liquid and gases is substantial. Both obey common lows of motion and both for most practical computational purposes can be regarded as continuum. Forces can be considered as body forces, example: forces due to gravity or they can be considered as surface forces example: surface tension.</p>
	<p>		<p>
	Based on speed and nature of flow, flows could be categorized into: <br>		Based on speed and nature of flow, flows could be categorized into: <br>

Zxaar at 06:06, 13 September 2005

2005-09-13T06:06:52Z

← Older revision		Revision as of 06:06, 13 September 2005
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	<p>		<p>
	Conservation laws can be applied to given quantity of fluid. However, it is more convenient to apply them on a small spatial region, called Control volume. For this purposes the geometry under consideration is subdivided into smaller sub-regions. Together they are called mesh or grid. And the approach used is called Control volume approach for solving the flow problem. <br>		Conservation laws can be applied to given quantity of fluid. However, it is more convenient to apply them on a small spatial region, called Control volume. For this purposes the geometry under consideration is subdivided into smaller sub-regions. Together they are called mesh or grid. And the approach used is called Control volume approach for solving the flow problem. <br>
-	The ~~figue~~ 1.1 shows one such computational domain. <br>	+	The figure 1.1 shows one such computational domain. <br>

	[[Image:Img_nm_intro_01.jpg]] <br>		[[Image:Img_nm_intro_01.jpg]] <br>

Zxaar at 05:27, 13 September 2005

2005-09-13T05:27:10Z

New page

Fluids could be regarded as substances those on application of external force deform or show very little resistance to deformation. This deformation in the cases of liquid and gases is substantial. Both obey common lows of motion and both for most practical computational purposes can be regarded as continuum. Forces can be considered as body forces, example: forces due to gravity or they can be considered as surface forces example: surface tension.

Based on speed and nature of flow, flows could be categorized into: 
*[[Laminar flow]] 
*[[Turbulent flow]] 
Or it could be categorized into: 
*[[Subsonic flow]] 
*[[Supersonic flow]] 
Based upon their [[Mach number]]. This Mach number could be used to determine whether the flow shall be considered as [[Incompressible flow ]] or [[Compressible flow]] for computational purposes. For the flows of Mach number smaller than 0.3, we can safely assume them as incompressible for all computational purposes. 

Based upon the degree of effect upon the viscosity on the flow, the flows could be classified as [[Viscous flow]] or [[Inviscid flow]] in nature. Fluids obeying [[Newton’s law]] are called [[Newtonian]].
Further based upon the phases of flow, the flows can be considered as [[multiphase]] or single phase flows. 

== Control Volume Approach ==

Conservation laws can be applied to given quantity of fluid. However, it is more convenient to apply them on a small spatial region, called Control volume. For this purposes the geometry under consideration is subdivided into smaller sub-regions. Together they are called mesh or grid. And the approach used is called Control volume approach for solving the flow problem. 
The figue 1.1 shows one such computational domain. 

[[Image:Img_nm_intro_01.jpg]] 
Figure 1.1 

== Conservation Principles ==
== Conservation of scalar quantities ==
== Simple Flows ==
=== [[Incompressible flow]] ===
=== [[Euler or inviscid flow]] ===
=== [[Potential flow]] ===
=== [[Stokes or creeping flow]] ===