|Mechanical and Aerospace Engineering, UC San Diego
Mechanical and Aerospace Engineering, UC San Diego
Engineering Mechanics, Zhejiang University
|Theoretical and numerical studies of differential equations and dynamical systems relevant to mixing and transport processes in environmental & geophysical flows and their impacts.|
|Arizona State University
Massachusetts Institute of Technology
UC San Diego
|Mathematical, Computational and Modeling Sciences Center
Mechanical & Aerospace Engineering, ASU
|Grants and Fellowships|
PI: "Collaborative Research: Theories and experiments on scalar mixing in chaotic flows", 2012-2015
|Fluid Stirring, Coherent Structures|
Fluid transport is the result of the interplay between stirring and diffusion processes. In nature additional chemical and physical processes impose further complexities to transport problems. In the past decade the use of dynamical systems methods has become popular to identify coherent structures as a result of stirring. Our group is interested in new applications that use information based on the identified coherent structures, and new models using measures from these coherent structures. Some on-going projects include identifying turbulence structures in atmospheric flows for aviation safety; pattern formation and modeling for pollutant transport; mixing dynamics for charged flows in earth's ionosphere; and modeling diffusion-reaction in ecological systems based on identification of the coherent structures. Overall the question that needs to be addressed is: given some resolved flow (from model or observations) what can we tell about the transport dynamics at different scales and what will that bring to our understandings of physical/chemical/biological/ecological processes in the environment that may have major impact on human activities?
|We live in an environment where turbulence is ubiquitous. With diffusion, turbulence redistributes scalars which irreversibly mix. To quantify irreversible mixing one is typically interested in the rate of mechanical energy dissipation and the decay rate of scalar variance. Our group is interested in using novel mathematical tools to estimate these quantities. One direction we have taken is the use of integral constraints and variational calculus to generate optimal bounds on energy dissipation and scalar variance. We use this approach to study irreversible mixing in environmental and geophysical flows. Recent projects include quantification of energy dissipation rate subject to novel physical constraints and optimal bound calculation on scalar variance. We are interested in generalizing these results to turbulent flows subject to different forcing mechanisms and seek the use of these quantifications in real parameterizations.|
|Internal gravity waves (IGW) is the stratified analog to surface waves. Energy is converted from the global flow to supply wave motion. It's significance in the ocean is such that after the generation at the site IGW can propagate away and break. This could lead to elevated mixing in the ocean interior, which helps homogenizing heat, salt and nutrient from different bodies of water. Experiments, simulations and analytical models have been developed to address the generation of IGW in the lab or in the ocean. Our group is interested in estimating the total rate of energy conversion and its modal composition of wave energy. Some past projects involve modeling the energy conversion rate given complex topography, and finding the energy pathways when IGW reflect at the boundaries. Our interest is to further develop theoretical and numerical tools for better models on IGW conversion in large-scale simulations.|
|Calculus For Engineers II||Syllabus|
Methods of integration, applications of calculus, elements of analytic geometry, improper integrals, Taylor series.
|Calculus For Engineers III||Syllabus|
Vector-valued functions of several variables, partial derivatives, multiple integration.
|MAT275(FA11)||Modern Differential Equations||Syllabus|
|Introduces differential equations, theoretical and practical solution techniques. Applications. Problem solving using MATLAB. Credit is allowed for only MAT 275 or 274 toward a mathematics degree.|
|Applied Linear Algebra||Syllabus|
Solving linear systems, matrices, determinants, vector spaces, bases, linear transformations, eigenvectors, norms, inner products, decompositions, applications. Problem solving using MATLAB.
|Intro Chaos/Nonlinear Dynamics||Syllabus|
Properties of nonlinear dynamical systems; dependence on initial conditions; strange attractors; period doubling; bifurcations; symbolic dynamics; Smale-Birkhoff theorem; and applications.
|Applied Partial Differential Equations||Syllabus|
Second-order partial differential equations, emphasizing Laplace, wave, and diffusion equations. Solutions by the methods of characteristics, separation of variables, and integral transforms.
|Differential Equations II||Syllabus|
Methods of characteristics. Classification of second order partial differential equations. Fourier Series, convergence theorems, Fourier transforms. Separation of variables. Greens identities. Fundamental solutions and Greens functions, sturm-Liouville problems. Harmonic Functions, maximum Principles.
|Applied Dynamical Systems Methods||Syllabus|
|Applies modern dynamical systems methods to fluid mechanics: bifurcations, normal forms, nonlinear dynamics, pattern formation, mixing, and Lagrangian chaos.|
|In the news|
|'Model simulates flow patterns of urban PM', Environmental Health Perspectives, NIH|
|'Urban wind flows deposit pollutants in repetitive patterns', Wired|
|'Wind Concentrates Pollutants with Unexpected Order in an Urban Environment', AIP Newswise|
|'The skeleton of water', The Economist|
|'Finding Order in the Apparent Chaos of Currents', New York Times|
|Research project highlights|
Dynamics of ion density and electric field subject to inertial gravity wave forcing during a wave breaking event.
Reference: The response of charged density due to breaking inertial gravity wave in the lower regions of ionosphere (2014), Tang, W. & Mahalov, A., Phys. Plasmas, accepted
|Variability of scalar reaction in coherent structures (right click to play flash movie)|
Varability of scalar reaction dependent on coherent structures. Reaction speed is enhanced in hyperbolic regions and suppressed in elliptic regions. The variability of the reaction rates can be modeled with finite-time Lyapunov exponents characterizing early-time separation of trajectories.
Reference: Dependence of advection-diffusion-reaction on flow coherent structures (2013), Tang, W. & Luna, C., Phys. Fluids, 25, 106602
|Stochastic coherent structures in realistic urban flows|
(Physics of fluids cover) Stochastic Lagrangian coherent structures in an urban flow based on realistic diffusivity. Signatures of structures still visible even when full scale of randomness near the atmospheric boundary layer is considered.
Reference: The geometry of inertial particle mixing in urban flows, from deterministic and random displacement models (2012), Tang, W., Knutson, B., Mahalov, A & Dimitrova, R., Phys. Fluids, 24, 063302
|Turbulent structures near Hong Kong International Airport (right click to play flash movie)|
Evolution of radial velocity detected by LIDAR along with backward-time FDFTLE extracted from the retrieved 2D velocity field. Also shown are Hovmoller diagrams at various ranges indicated by the dotted lines in the movies. First movie: recirculation bubble detachment and regeneration next to a mountain peak. Second movie: a persistent ridge of updraft originated inside a mountain gap.
Reference: Lagrangian Coherent Structure analysis of terminal winds detected by LIDAR. Part II: structure evolution and flight data analyses (2011). Tang, W. Haller, G. & Chan, P.W., J. Appl. Meteorol. Clim., 50, 2167-2183
|Lagrangian signatures of a jet stream and balloon measurements|
Comparison between the repelling LCS generated from model data and atmospheric characteristics estimated by weather balloon measurements. Color contour is the Forward-time DLE. Black line originating from the big island is the balloon trajectory, green curve is refractive index structure constant Cn2 and blue curve is the dissipation rate.
Reference: Lagrangian Coherent Structures Near a Subtropical Jet Stream (2010), Tang, W., Mathur, M., Haller, G., Hahn, D.C. & Ruggiero, F.H., J. Atmos. Sci., 67, 7, 2307-2319.
|Finite-size Pollutant Particle Transport in Urban Street Canyon (right click to play flash movie)|
Attracting Lagrangian Coherent Structures (LCS) inside an urban street canyon for finite-size inertial particles. The LCS are computed from the slow manifoid velocity derived from simulation data. The forward-time motion shows that an inertial particles is attracted to the local maxima of the Direct Lyapunov Exponent field. The backward-time motion shows the results for inversion of the finite-size particle using different schemes.
Reference: Locating an atmospheric contamination source using slow manifolds (2009), Tang, W., Haller, G., Baik, J.-J. & Ryu, Y.-H., Phys. Fluids, 21, 043302.
|Baik, J.-J. Seoul National University|
|Calhoun, R. MAE, ASU|
|Caulfield, C.P. DAMTP, University of Cambridge|
|Chan, P.W. Hong Kong Observatory|
|del-Castillo-Negrete, D., Oak Ridge National Lab|
|Dimitrova, R., CEE, Notre Dame|
|Fernando, H.J.S., CEE, Notre Dame|
|Hahn, D.C. Hanscom AFB|
|Haller, G. ETC Zurich|
|Kerswell, R.R. Math, University of Bristol|
|Llewellyn Smith, S.G. MAE, UC San Diego|
|Ouellette, N, ME, Yale|
|Mahalov, A. Math, ASU|
|Peacock, T. ME, MIT|
|Ruggiero, F.H. Hanscom AFB|
|Stocker, R. CEE, MIT|
|Taylor, J.R. DAMTP, University of Cambridge|
|Thiffeault, J.-L., Mathematics, Wisconsin|
|Young, W.R. SIO|
|Brent Knutson, PhD Candidate, Applied Mathematics||James Upton, Undergraduate, Mathematics|
|Phillip Walker, PhD Candidate, Applied Mathematics||Craig Gassaway, Undergraduate, Math & Physics|
|Kimberly Jones, MSc, Applied Mathematics||Christopher Luna, Undergraduate, Math & Physics|
|Aditya Dhumuntarao, Undergraduate, Math & Physics|
|Tim Lai, Undergraduate, Mathematics||Angelica Deibel, Undergraduate, Mathematics|
|Christian Wake, Undergraduate, Mathematics||Hershey Kelley, Undergraduate, Mathematics|
|Juan Durazo, Undergraduate, Mathematics|
|Inez Ibarra, Biosciences High School||Lena Tamrat, Biosciences High School|
|Refereed Journal Publications|
|1.||Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in a plane stratified Couette flow (2004) Caulfield, C.P., Tang, W. & Plasting, S.C., J. Fluid Mech. 498, 315-332. [pdf] DOI: 10.1017/S0022112003006797|
|2.||Bounds on dissipation in stress-driven flow (2004) Tang, W., Caulfield, C.P. & Young, W.R., J. Fluid Mech. 510, 333-352. [pdf] DOI: 10.1017/S0022112004009589|
|3.||Bounds on dissipation in stress-driven flow in a rotating frame (2005) Tang, W., Caulfield, C.P. & Young, W.R., J. Fluid Mech. 540, 373-391. [pdf] DOI:10.1017/S0022112005005926|
|4.||Locating an atmospheric contamination source using slow manifolds (2009), Tang, W., Haller, G., Baik, J.-J. & Ryu, Y.-H., Phys. Fluids, 21, 043302. [pdf] URL: http://link.aip.org/link/?PHF/21/043302 DOI: 10.1063/1.3115065|
|5.||A prediction for the optimal stratification for turbulent mixing (2009), Tang, W., Caulfield, C.P. & Kerswell, R.R., J. Fluid Mech., 634, 487-497. [pdf] DOI:10.1017/S0022112009990711|
|6.||Accurate extraction of LCS over finite domains, with applications to flight data analyses over Hong Kong International Airport (2010), Tang, W., Chan, P.W. & Haller, G., Chaos, 20, 017502. [pdf] DOI: 10.1063/1.3276061|
|7.||Lagrangian Coherent Structures and internal wave attractors (2010), Tang, W., Peacock, T., Chaos, 20, 017508. [pdf] DOI:10.1063/1.3273054|
|8.||Lagrangian Coherent Structures near a subtropical jet stream (2010), Tang, W., Mathur, M., Haller, G., Hahn, D.C., Ruggiero, F.H., J. Atmos. Sci.,67, 7, 2307-2319[pdf] DOI: 10.1175/2010JAS3176.1|
|9.||Lagrangian dynamics in stochastic inertia-gravity waves (2010), Tang, W., Taylor, J.E. & Mahalov, A., Phys. Fluids, 22,126601. [pdf] doi:10.1063/1.3518137|
|10.||Lagrangian Coherent Structure analysis of terminal winds detected by LIDAR. Part I: turbulence structures (2011), Tang, W., Haller, G. & Chan, P.W., J. Appl. Meteor. Clim., 50, 325-338[pdf] DOI:10.1175/2010JAMC2508.1|
|11.||Lagrangian Coherent Structure analysis of terminal winds detected by LIDAR. Part II: structure evolution and flight data analyses (2011), Tang, W., Haller, G. & Chan, P.W., J. Appl. Meteor. Clim., 50, 2167-2183[pdf] DOI:10.1175/2011JAMC2689.1|
|12.||The geometry of inertial particle mixing in urban flows, from deterministic and random displacement models (2012), Tang, W., Knutson, B., Mahalov, A & Dimitrova, R., Phys. Fluids, 24, 063302. [pdf] DOI:10.1063/1.4729453 Phys. Fluids cover|
|13.||Finite-time statistics of scalar diffusion in Lagrangian coherent structures (2012), Tang, W. & Walker, P., Phys. Rev. E, 86, 045201(R).[pdf] DOI: 10.1103/PhysRevE.86.045201PRE Kaleidoscope|
|14.||Stochastic Lagrangian dynamics for charged flows in the E-F regions of ionosphere (2013), Tang, W. & Mahalov, A., Phys. Plasmas, 20, 032305, [pdf] DOI:10.1063/1.4794735|
|15.||Dependence of advection-diffusion-reaction on flow coherent structures (2013), Tang, W. & Luna, C., Phys. Fluids, 25, 106602, [pdf] DOI: 10.1063/1.4823991|
|16.||The response of charged density due to breaking inertial gravity wave in the lower regions of ionosphere (2014), Tang, W. & Mahalov, A., Phys. Plasmas, accepted [pdf]|
|1.||Lagrangian Coherent Structure Analysis of Terminal Winds: three-dimensionality, intra-model variations, and flight analyses, Knutson, B., Tang, W., Chan, P. W. (for J.Appl. Meteor. Clim.)|
|To be submitted|
|Concept based learning workshop for high school students|
This workshop is aimed at engaging high-achieving high school students with continued motivation for further mathematical studies. Mathematical models are introduced along with development of physical concepts so the students will learn both simultaneously. Various types of ODE models for natural systems are formed and the concept of dynamics is developed through discussions of the meaning of the models. Students in this workshop get hands-on experience in applying mathematical and numerical tools to analyze the dynamical behaviors of the models. Topics such as the Taylor series expansion, vector calculus, linear algebra, differential equations and phase portraits, suitable to the problems presented, are connected so they will not be perceived as disjointed math subjects in future studies. MATLAB is used for numerically modeling the dynamical systems.
High school students in this workshop also get the opportunity to meet with successful undergraduate students in applied mathematics on experiences and tips being a math and science major. Interactions with college students and faculty help the participating students learn from role models and be informed with career opportunities in STEM fields. Past participants attend UC Berkeley, UCLA, U Chicago, Harvey Mudd College and the Barrett Honors College at ASU.
A presentation on applied dynamics can be located here.
Workshop notes can be located here.
|Workshop Participants, 2011||Workshop Participants, 2012|
|Workshop Participants, 2013|
|Applied Math internship for high school students|
High school students can also participate in research activities at our lab. Throughout the course of the internship the students will be exposed to different mathematical concepts linked to solve applied problems in different disciplines. Concept based learning of mathematical topics is further solidified through hands on research questions.
Currently students from BioSciences High School, Phoenix have benefitted from participating in this internship.