Scale invariance and self‐similarity in kinematic wave overland flow in space and time

Haltaş, İsmail; Kavvas, M. L.

doi:10.1002/hyp.8092

Cited by 8 publications

(5 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another good applied reference is Olver (1986). The Lie group method is used to investigate the scale invariance of kinematic wave overland flow problem in Haltas and Kavvas (2011b). An algorithm developed for finding symmetry transformations using the group structure has been discussed in Cayar and Kavvas (2009a) and applied by engineers in Cayar and Kavvas (2009b) to find symmetries in a heterogeneous unconfined aquifer problem.…”

Section: Introductionmentioning

confidence: 99%

A comparison of the modern Lie scaling method to classical scaling techniques

Polsinelli¹,

Kavvas²

2016

Hydrol. Earth Syst. Sci.

View full text Add to dashboard Cite

Abstract. In the past 2 decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics, including but not limited to overland flow, groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non-dimensionalization, and the Vaschy-Buckingham-theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is made with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables, which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.

show abstract

Section: Introductionmentioning

confidence: 99%

A comparison of the modern Lie scaling method to classical scaling techniques

Polsinelli¹,

Kavvas²

2016

Hydrol. Earth Syst. Sci.

View full text Add to dashboard Cite

show abstract

“…Lie groups and symmetries and their applications to differential equations were discussed by Hansen (1964), Bluman and Cole (1974), Ibragimov (1994;1995), and Bluman and Anco (2002). Recently, oneparameter Lie group of point scaling transformations were utilized to investigate scale invariance and self similarity of various hydrologic processes (Haltas and Kavvas 2011a;2011b), one dimensional open channel flow process (Ercan et al, 2014), and one dimensional suspended sediment transport process (Carr et al, 2015). One of the main advantages of this methodology is that the self-similarity conditions due to the initial and boundary conditions can also be investigated in addition to the conditions due to the governing equations.…”

Section: Introductionmentioning

confidence: 99%

Scaling and self-similarity in two-dimensional hydrodynamics

Ercan

Kavvas

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

The conditions under which depth-averaged two-dimensional (2D) hydrodynamic equations system as an initial-boundary value problem (IBVP) becomes self-similar are investigated by utilizing one-parameter Lie group of point scaling transformations. Self-similarity conditions due to the 2D k-ε turbulence model are also investigated. The self-similarity conditions for the depth-averaged 2D hydrodynamics are found for the flow variables including the time, the longitudinal length, the transverse length, the water depth, the flow velocities in x- and y-directions, the bed shear stresses in x- and y-directions, the bed shear velocity, the Manning's roughness coefficient, the kinematic viscosity of the fluid, the eddy viscosity, the turbulent kinetic energy, the turbulent dissipation, and the production and the source terms in the k-ε model. By the numerical simulations, it is shown that the IBVP of depth-averaged 2D hydrodynamic flow process in a prototype domain can be self-similar with that of a scaled domain. In fact, by changing the scaling parameter and the scaling exponents of the length dimensions, one can obtain several different scaled domains. The proposed scaling relations obtained by the Lie group scaling approach may provide additional spatial, temporal, and economical flexibility in setting up physical hydraulic models in which two-dimensional flow components are important.

show abstract

“…More recently, this was related to Lie groups of symmetry. [22][23][24] On the other hand, another approach was based on stochastic differential calculus. [25][26][27] In a given way, this paper combines both approaches with the help of the complementary properties of stable L evy processes and Clifford algebra.…”

Section: Introduction a What Is At Stake?mentioning

confidence: 99%

Multifractal vector fields and stochastic Clifford algebra

Schertzer

Tchiguirinskaia

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

A nonlinear approach to transition in subcritical plasmas with sheared flow Physics of Plasmas 24, 122307 (2017) In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable L evy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of L evy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the L evy stability grants a given statistical universality. The multifractal framework is very convenient to analyze and simulate extremely variable fields over a wide range of space-time scales. However, it has been mostly developed for scalar fields and is therefore not yet suitable for many applications. This paper demonstrates that combining two special, but wide enough classes of stochastic processes ("stable Levy processes") and algebra ("Clifford algebra") provides a convenient mathematical and physical framework for vector multifractals and possibly manifold valued multifractals, i.e., for multifractal fields whose values belong to a manifold (e.g., a flow on this manifold), therefore for a wide range of complex systems. This paper does not intend to be reserved for specialists and therefore provides pedagogical introductions to concepts that are used.

show abstract

Scale invariance and self‐similarity in kinematic wave overland flow in space and time

Cited by 8 publications

References 17 publications

A comparison of the modern Lie scaling method to classical scaling techniques

A comparison of the modern Lie scaling method to classical scaling techniques

Scaling and self-similarity in two-dimensional hydrodynamics

Multifractal vector fields and stochastic Clifford algebra

Contact Info

Product

Resources

About