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Department of Mathematics and Statistics

Jahrul Alam

Associate Professor of Mathematics
PhD, McMaster
Computational Fluid Dynamics

Office: HH-3054
Phone: (709) 864-8071
Fax: (709) 737-3010


Research Projects

A) Wavelet-based turbulence modelling and simulation

B) Data-driven turbulence modelling and simulation

Wavelet-based multi-scale analysis of a turbulent velocity u'(t)

What are wavelets? Wavelets are a particular type of functions that are powerful mathematical tool for understanding large-scale data which possess coherent structures over a random background. Wavelet analysis transform a given data at different scales or resolution, thereby leading to a reduced-order representation of turbulent flow.

The basic idea of multi-scale wavelet analysis for atmospheric turbulence is illustrated here with an example. For the velocity u'(t) shown in the first row, the dominant frequency is 100 [mHz], which is 10 [mHz] for the velocity in the second row. The classical Fourier-based energy spectrum is shown in the first column. It is observed that the an evident peak energy appears at frequency of 100 [mHz], which is exactly detected by the Fast Fourier Transform.

The multiscale time-frequency energy spectrum is shown in the second column. It is clearly observed that the strongest positive peak appears at 100 [mHz] and 10 [mHz], respectively, for the entire duration of time.

Data-driven turbulence modelling: This approach utilizes a machine learning algorithm. Consider that a collocation of turbulence data exists from satellite observation of weather, from an experiment, or from an LES run. Proper orthogonal decomposition is one of the most important techniques for data-driven turbulence modelling in which the computer learns from a given data in order to predict turbulence. To illustrate the basic idea, I have generated a snapshot matrix 262144 x 100, the last column of two such data are visualized in the first column of the following illustration.

The last data of the snapshot matrix First 4 modes of the POD basis

The second column shows 4 modes of the POD basis, which are assocated to the largest singular value of the given data. In the first row, it can be seen that first two POD modes can capture the dominant feature of the data. In the second row, the first POD mode captures the dominant features of the data.

This example tells us that if we have some data for a turbulent flow, we can apply simple numerical techniques to extract dominant patterns of turbulence, which can be used for predicting turbulence. I am looking for potential graduate students working on this project.

Wavelet-based reduced order turbulence modelling algorithms: The wavelet-based turbulence modelling is similar, in principle, to the POD method. Here, we project the turbulent flow on a wavelet-basis, and consider the most significant wavelet modes. In the following illustration, I have simulating two-dimensional turbulence, which is done by considering only 1% of the wavelet-modes.

decaying turbulence forced turbulence
decaying turbulence forced turbulence

Figure: The vorticity fields at 128 eddy turn over times, showing intermittency of two-dimensional turbulence at moderate Reynolds number. Left: decaying turbulence, right: forced turbulence.

Modeling gray-zone turbulence with robust identification of coherent flow structures

Let us start by considering homogeneous isotropic turbulence. I am interested in robust methods for identifying dynamically distinct regions in turbulent flow, and using such flow structures for modeling dissipation in atmospheric turbulence. Atmospheric turbulence is characterized by a 'gray-zone' in which dynamically distinct flow regime interact without exhibiting a clear flow seperation. As a result, traditional large eddy simulation (LES) requires an extremely large number of computational degrees of freedom.

Coherent flow structure Rate of turbulence dissipation

Symmetry- and dissipation-preserving wavelet-based JFNK methodology

Accurately capturing convective fluxes in numerical simulations is a long standing problem. Attempts to conserve convective fluxes with higher order methods would violate the maximum principles of the governing partial differential equations. Nonlinear schemes are promising candidates for obtaining higher order wiggle free simulations results. Such schemes can be linearized in time so that multigrid or Krylov methods can be applied, or nonlinear multigrid full approximation scheme (FAS) can be applied without linearization.

Many flux limited (van Leer) schemes of positive type do not preserve the skew-symmetry of the convective operators. The numerical dissipation considered by such schemes is not consistent with the subgrid scale dissipation considered by LES. I am developing symmetry preserving wavelet-based discretization techniques, where the resulting nonlinear system is solved by the Jacobian-free Newton-Krylov method.

In Computational Fluid Dynamics (CFD), there is a growing interest in developing high-fidelity simulation tools for atmospheric turbulence, wind energy projects, enhance oil and gas recovery, dispersion of pollutants in cities, boundary layer meteorology, to name a few. Over the past decades, a number of algorithms for velocity-pressure coupling was derived for fluid flow simulations. Nevertheless, development of more appropriate numerical methodology for the Navier-Stokes equations is necessary to avoid significant modelling and simulation errors.

A key feature of the wavelet-based numerical method is that the it results into a well-conditioned discretization. It then employs Newton's method to solve the nonlinear system resulted from 'Beam and Warmming' type time integration scheme, where a Krylov subspace iterative method adopted to invert the Jacobian. It encompasses essential features of the JFNK (Jacobian-free Newton-Krylov) method, while wavelets provide multi-resolution discretization.

The left figure is the exact pressure field, and the right is figure the simulated pressure field, using a mesh of 512 x 512 collocation points. The mesh was varied like 65 x 65, 128 x 128, 256 x 256, 512 x 512. It was observed that the number of Krylov iterations was about the same at each grid, which confirms that the Jacobian is well-conditioned.

Exact solution Numerical solution

The wavelet-based JFNK algorithm was tested for decaying and forced isotropic turbulence, as shown below.
decaying turbulence forced turbulence
decaying turbulence forced turbulence

Figure: The vorticity fields at 128 eddy turn over times, showing intermittency of two-dimensional turbulence at moderate Reynolds number. Left: decaying turbulence, right: forced turbulence.














Multiscale large eddy simulation (LES) and sub-grid scale numerical modeling of atmospheric boundary layer turbulence

Spatial and global intermittency of turbulence is a phenomena in the atmospheric boundary layer turbulence, as well as in other geophysical flows, which is poorly understood. Accurate numerical prediction of intermittent turbulent bursts is important for accurate projection of climate change.

For example, to accurately interpret the exchange of CO2 between the atmosphere and forests through tower measurements of CO2 fluxes, it is necessary to understand mixing and transport by winds within and above forest canopies over complex terrain. More specifically, a substantial fluxes of CO2 carried by intermittent turbulence generated by drainage flows of the nighttime boundary layer may not be measured by the towers. In other words, current state-of-the art technology for predicting global warming trend calls for new research on the atmospheric boundary layer.

The lowest region of the atmospheric boundary layer is where we live and breath. The air at the ground level stops flowing due to friction. If we move upward from the ground, the wind speed gets faster and faster, as well as wind vector rotates. However, a knowledge of wind vectors is insufficient to interpret tower measurements without quantifying the degree of turbulence experienced at a measurement site. Moreover, when the ground is heated by the sun at daytime, the air above the ground becomes hot leading to a turbulent mixing zone. When the ground is cooled at nighttime, turbulence above the ground is often characterized by a brief episode of bursting phenomena.

Explaining this very complex phenomena of the atmospheric boundary layer requires knowledge of geophysical fluid dynamics and high performance scientific computing. It is difficult - if not impossible - to directly calculate turbulent eddies at a length scale O(1 m) because such eddies interact in a region that extends to meteorological scales, horizontally O(100 km) and vertically O(10 km). When the numerical resolution approaches the turbulent sub-grid scale, meteorological modeling techniques become inappropriate. Similarly, sub-grid scale turbulence models also become inappropriate, when the length scale approaches to the meteorological scale. This challenge is often called Terra-Incognita in the area of atmospheric turbulence.

Project 1:
Canopy stress based numerical models may provide energetic feedback to sub-grid scale turbulent models, particularly to avoid a fine numerical grid near the ground where turbulent eddies are anisotropic. One aim of this project is to quantify mixing and transport above complex terrain in the presence/absence of forests. In other words, how does a destabilization of the energy flux at a height 200m to 500m above the ground leads to daytime cooling and nighttime warming?


Project 2:
Current trend in global warming due to burning fossil fuel may suggests Governments to promote clean energy projects. Although some climate models predict that making 10% of the global energy demand from wind energy may warm the atmosphere by another 1C by the end of next century, debates continue whether wind farms actually warms the atmosphere or re-distribute energy. One aim of this project is to develop a computational model for 'wind farm turbulence' for the Terra-Incognita flow regime.

Project 3:
The advancement of adaptive wavelet methodology provides a natural framework to filter the turbulent eddies into a group of significant eddies which maybe resolved by an adaptive mesh, and a group of non-significant eddies, the cascade of which maybe parameterized. One aim of this project is to understand a sophisticated parametization methodology when significant eddies are filtered with the second generation wavelet transform.

Research areas

  • Multiscale wavelet method
    I am looking at multiscale computation of the solution of nonlinear partial differential equations using second generation wavelets. This is a novel multiscale modelling approach for solving problems, having multiscale features.
  • Turbulence
    I am interesting in modelling intermittency of turbulence in the atmospheric boundary layer, where we live and breadth. My recent interest is on the development of a multiscale eddy simulation (MES) model, where significant eddies are tagged with wavelets, and the resulting filtered flow is solved with a classical LES model as well as an implicit LES model, where the implicit LES approach is novel idea within the MES modelling approach.
  • Geophysicsl fluid dynamics
    Although Geophysical Fluid Dynamics commonly points to large scale flow problems related to the atmosphere and ocean, I am mainly interested in subsurface flows in a saline aquifer or in an oil reservoir. Main research focus is on multiscale modelling of flow and transport in porous media using the volume averaging method. For this class of problems, I am mainly investigating the development of a multiscale modelling approach for flows with fingeing instabilities.

    Opportunities

    I am open to discuss on possible projects for PhD, MSc, and undergraduate students. As a research assistant, one must be an expert programmer in C++. Depending on the topics, students work with multigrid methods and Lagrangian methods. Writing new code and verification of multiscale model are among responsibilities of graduate assistants.

Objectives

A better understanding of fluid flows that are important in environmental, aeronautical, or industrial applications requires improving our current knowledge of turbulence. For example, accurate modelling of the turbulent atmosphere is critical for such varied purposes as weather forecasting, projecting climate change, and mitigating air pollution.

The development and the verification of high performance adaptive multiresolution models are key objectives of my current research. Following are some specific on-going research topics.


Multi-scale, space-time adaptive algorithms for turbulent flows


Turbulence is difficult to approximate mathematically, and to calculate numerically, because it is active over a large and continuous range of length scales (e.g. from less than a millimeter to hundreds of kilometers in the atmosphere). The range of active scales increases with an increase of the Reynolds number, which means flows are increasingly difficult to calculate at large Reynolds numbers of practical interest. However, it has been conjectured that a turbulent flow is spotty - only a fraction of the flow is active and this active proportion of the flow decreases as the Reynolds number increases. This means that high Reynolds number flows are highly intermittent in both space and time. A numerical model that exploits such space-time intermittency would use only a fraction of computational time compared to classical high performance numerical models.

A better understanding for the space-time intermittency of turbulence is related to many environmental or industrial applications, but classical theories or models of turbulence fail to explain properly such intermittency. In particular, a scaling of the intermittent space-time degrees of freedom with respect to increasing turbulence intensity would help us in designing high performance computer models. In my PhD thesis, a scaling of the number of space-time intermittent modes with the Reynolds number for 2D homogeneous isotropic decaying turbulence was estimated numerically. This study further reported that temporal intermittency is much stronger than the spatial intermittency for 2D decaying turbulence.

Currently, I am involved in extending these results to 3D models of turbulence. I am also investigating for a scaling of intermittent space-time modes in the case of forced homogeneous isotropic turbulence.


Coherent structures in the atmosphere


Our current knowledge of coherent motion in the atmosphere relies on adhoc approximation of the average motion, which is an accumulated empirical or statistical information. An improved understanding of the coherent atmosphere is essential for projecting climate change or improving global climate models. State-of-the-art computer models for the atmosphere attempt to resolve a flow up to a certain scale, expressing unresolved scales in terms of resolved motion. In such subgrid scale approaches, the intermittency of coherent structures are ignored. However, it is evident from both numerical simulation and observation that only a fraction of the turbulent atmospheric scales are needed to be resolved to exploit intermittency. Therefore, exploiting intermittency is an optimal alternative to classical subgrid scale modelling. Until recently, it is not yet clear how does one extract intermittently active atmospheric scales. I study the space-time intermittency of highly turbulent flows i.e. flows with high Reynolds number.

This work aims to develop multiresolution approaches for investigating intermittency of coherent structures in the atmosphere. A multiresolution atmospheric modelling system has been proposed and verified using a coastal circulation system of a dry atmosphere. Further extension to understand an appropriate parameterization for moisture effect and turbulence are in progress.


Fully-Lagrangian adevection schemes for industrial, environmental, or geoscience applications


Lagrangian Upwind
Figure: Numerical simulation of a moving front in a channel. Top: Fully-Lagrangian, bottom: Eulerian.

A computer model of the atmosphere or ocean concerns advection dominated flows. The realization that numerical treatment of advection on a conventional Eulerian mesh is plagued with instabilities and unrealistic negative constituent values has inspired continuous efforts in finding more elegant tools for improving state-of-the-art atmospheric transport and chemistry models.

A fully-Lagrangian advection scheme has been developed for accurate simulation of advection dominated flow problems. The model has been compared with standard Eulerian finite different approaches. We found that the fully-Lagrangian model provieded significant improvements in terms of both CPU time and accuracy. Two types of problems were considered: a two-dimensional flow, where a fluid is injected into a domain confined in one direction and containing a resident fluid, and a two-dimensional sea-breeze circulation of a dry atmosphere in the coastal region.

A computer model for the geological storage of greenhouse gasses, and for oil/gas reservoir simulation are potential application of this method. In addition, I am also interested in extending this work towards a fully-Lagrangian 3D simulation of the atmosphere, and in comparing the result with classical approaches - for example - semi-Lagrangian and flux-form Eulerial schemes.


Energy-conserving Computational Fluid Dynamics (CFD) techniques in complex geometry


In aerodynamics, off-shore drilling, or wind engineering of buildings, one needs simulate moderate to high Reynolds number incompressible flows around arbitrary solid structures. For an adaptive mesh simulation of such flows, a potential challenge is to resolve the coupling between the velocity and pressure such that the incompressibility of the flow is satisfied.

I study the development of novel energy-conserving algorithms for flow around arbitrary obstacles. In this approach a penalization method is used to model both the pressure gradient force and the force exerted by solid obstacles. I am interested to examine this algorithm for complex geometry flows and to compare the result with that of classical projection algorithms.

vortex shedding grid

Figure: Vortex shedding at the wake behind a cylinder. Left: Adaptive wavelet solution, right: Adapted grid