Order of reaction: Difference between revisions
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{{For|kinematic waves in traffic flow|Traffic flow}} | |||
In gravity and pressure driven fluid dynamical and geophysical mass flows such as ocean waves, avalanches, debris flows, mud flows, flash floods, etc., kinematic waves are important mathematical tools to understand the basic features of the associated wave phenomena. | |||
<ref name=takahashi_07>{{cite book | |||
|last=Takahashi |first=T. | |||
|year=2007 | |||
|title=Debris Flow: Mechanics, Prediction and Countermeasures | |||
|publisher=[[Taylor and Francis, Leiden]] | |||
}} | |||
</ref> | |||
These waves are also applied to model the motion of highway [[traffic flows]].<ref name=lighthill_whitham_1955>{{cite journal |last=Lighthill |first=M.J.|last2=Whitham |first2=G.B. |year= 1955 |title=On kinematic waves. I: Flood movement in long rivers. II: A theory of traffic flow on long crowded roads | |||
|journal=[[Proceedings of the Royal Society]] | |||
|volume=229A |issue=4 |pages=281–345 | |||
}} | |||
</ref> | |||
<ref name=newell_1993>{{cite journal |last=Newell |first=G.F. |year= 1993 |title=A simplified theory of kinematic waves in highway traffic, Part I: General theory | |||
|journal=[[Transpn. Res. B]] | |||
|volume=27B |issue=4 |pages=281–287 | |||
}} | |||
</ref> | |||
In these flows, mass and momentum equations can be combined to yield a kinematic wave equation. Depending on the flow configurations, the kinematic wave can be linear or non-linear, which depends on whether the wave celerity is a constant or a variable. Kinematic wave can be described by a simple [[partial differential equation]] with a single unknown field variable (e.g., the flow or wave height, <math>h</math>) in terms of the two independent variables, namely the time (<math>t</math>) and the space (<math>x</math>) with some parameters (coefficients) containing information about the physics and geometry of the flow. In general, the wave can be advecting and diffusing. However, in simple situation, the kinematic wave is mainly advecting. | |||
== Kinematic wave for debris flow == | |||
Non-linear kinematic wave for debris flow can be written as follows with complex non-linear coefficients: | |||
:<math>\frac{\partial h}{\partial t} + C \frac{\partial h}{\partial x} = D \frac{\partial^2 h}{\partial x^2},</math> | |||
where <math>h</math> is the debris flow height, <math>t</math> is the time, <math>x</math> is the downstream channel position, <math>C</math> is the pressure gradient and the depth dependent nonlinear variable | |||
wave speed, and <math>D</math> is a flow height and pressure gradient dependent variable diffusion term. | |||
This equation can also be written in the [[conservative form]]: | |||
:<math>\frac{\partial h}{\partial t} + \frac{\partial F}{\partial x} =0,</math> | |||
where <math>F</math> is the generalized flux that depends on several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient. For <math>F=h^2/2</math>, this equation reduces to the [[Burgers' equation]]. | |||
==References== | |||
<references/> | |||
{{physical oceanography}} | |||
[[Category:Fluid dynamics]] | |||
Revision as of 05:32, 27 January 2014
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.
In gravity and pressure driven fluid dynamical and geophysical mass flows such as ocean waves, avalanches, debris flows, mud flows, flash floods, etc., kinematic waves are important mathematical tools to understand the basic features of the associated wave phenomena. [1] These waves are also applied to model the motion of highway traffic flows.[2] [3]
In these flows, mass and momentum equations can be combined to yield a kinematic wave equation. Depending on the flow configurations, the kinematic wave can be linear or non-linear, which depends on whether the wave celerity is a constant or a variable. Kinematic wave can be described by a simple partial differential equation with a single unknown field variable (e.g., the flow or wave height, ) in terms of the two independent variables, namely the time () and the space () with some parameters (coefficients) containing information about the physics and geometry of the flow. In general, the wave can be advecting and diffusing. However, in simple situation, the kinematic wave is mainly advecting.
Kinematic wave for debris flow
Non-linear kinematic wave for debris flow can be written as follows with complex non-linear coefficients:
where is the debris flow height, is the time, is the downstream channel position, is the pressure gradient and the depth dependent nonlinear variable wave speed, and is a flow height and pressure gradient dependent variable diffusion term. This equation can also be written in the conservative form:
where is the generalized flux that depends on several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient. For , this equation reduces to the Burgers' equation.
References
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
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