This effect is particularly important for structures on the boundary of different topographical features, such as those in Chicago, which sits next to Lake Michigan and thus feels the brunt of wind sweeping in from across the surface of the water while at the same time feeling the effects of the polar jet stream, or Los Angeles, which is positioned between the ocean and a range of tall mountains and so experiences the turbulence of ocean winds hitting the land. Furthermore, because the acceleration or deceleration of the velocity profile diffuses itself through turbulence, the higher up a structure, the more turbulence it will experience (Azad 1993). Computational Fluid Dynamics

Having provided an overview of the atmospheric boundary layer and the areas of the ABL that most influence wind effects on high-rise buildings, it will now be possible to discuss computational fluid dynamics in greater detail in order to demonstrate how one might use numerical modeling in order to measure the wind excitations of any given design. Put simply, numerical modeling uses computers capable of rapidly performing millions of calculations in order to build models of the complex movements of fluids, and in this case, air. In general there is a trade-off one must make when using numerical modeling, because although there are a wide variety of equations and simulations possible, in most cases one must strike a balance between simplicity, accuracy, and speed.

The simplest method available for modeling flow are simple linear models, which have the benefit of simplicity and speed but which are ultimately insufficient for the kind of modeling needed to determine the ideal high-rise cross-sections. In contrast, direct numerical simulation, in which a computer simulates the Navier-Stokes equations "for a full range of turbulent motions for all scales," offers stunning accuracy and completeness, such that "when properly carried out, DNS results would be comparable in every way to quality experimental data" (Stangroom 2004, p. 74). This is because direct numerical simulation allows one to clearly define every variable and thus receive insight into each element of a flow pattern. However, the major drawback of direct numerical simulation is the sheer amount of processing power it requires; "as an example, high Reynolds number flows with complex geometries could require the generation of 1020 numbers," and even if engineers had access to such potent computing equipment, there is still not a guarantee that this would produce satisfactory results (Stangroom 2004, pp. 74-75). Thus, while direct numerical simulation holds great potential for the near future, when the extreme processing power required should become cheaper and more ubiquitous, in the mean time it is mostly used for smaller-scale modeling of flows with low Reynolds numbers.

Until direct numerical simulation of flows at high Reynolds numbers becomes practical, Large Eddy Simulation or LES has been shown to serve as a suitable replacement. LES has allowed researchers to effectively model a number of complex flows and accurately predict certain forms of turbulence, particularly in regards to the effect of surface fluctuations on turbulence (Stangroom 2004, pp. 76-76). LES has been demonstrated to be more accurate than other kinds of modeling for certain situations, and particularly when predicting turbulence, but it still carries some computational requirements that may make it a less attractive option. Nevertheless, LES has proven a useful tool where other simulations are either too simple or too complex to reasonably use.

Arguably the best modeling currently available comes in the form of Reynolds Averaged Navier-Stokes (RANS) equations, non-linear equations which solved the initial problem that the Navier-Stokes equations were really only applicable to laminar flows and not turbulent ones (Stangroom 2004, p. 32). Furthermore, RANS modeling is far less costly than either direct numerical simulation and LES, and it is performable using widely available commercial software, rather than specially-designed or contracted computers and equipment. While RANS models are nowhere near as accurate as direct numerical simulation and somewhat less accurate than LES in certain situations, for most applications it makes up for these limitations due to its speed and ease of use. Furthermore, for certain simulations researchers have proposed a detached eddy simulation, in which "the whole boundary layer is modeled using a RANS model and only separated regions (detached eddies) are modeled by LES" (Stangroom 2004, p. 77). This allows one to benefit from the greater accuracy of LES where important but not spend undue computational resources on attempting to model the entire boundary layer via LES.

Turbulence modeling

Arguably the most complex area of computational...

At its simplest, turbulence, or at least the movement of turbulence, might be described as "the whole cascade of energy down through smaller and smaller scales until finally a limit is reached when the eddies become so tiny that viscosity takes over," and this description reveals some of the difficulties related to modeling turbulence. For one, it is extremely difficult to model each scale of the entire process even as the movement between these scales is the core of what is being examined. Furthermore, when using RANS to simulate turbulent flows one must draw on additional models, because the formulation of the RANS equations leaves the set not closed; this is one of the major distinctions between direct numerical simulation, which deals with the directly solvable Navier-Stokes equations, and the RANS equations, which require additional equations in order to effectively solve the remaining unknown terms, dubbed Reynolds stresses (Stangroom 2004, p. 79-80).
Though a number of additional models have been developed to help model turbulence, the first "industry standard" is the k-? (k-epsilon) model, which uses two equations in order to model the energy dissipation per unit mass and the kinematic viscosity (Stangroom 2004, p. 83). In the model k is the kinetic energy per unit mass and ? is the dissipation rate of kinetic energy as heat by the action of viscosity, which allows one to define both the velocity and length scales at any given time and space. With this information in hand, one can then define eddy viscosity which allows one to then make k and ? The subject of transport equations.

The k-? model has become the standard for modeling turbulence largely because "it has relatively low computational costs and is numerically more stable than the more advanced and complex stress models," and "has been verified and validated for a wide variety of flows" (Stangroom 2004, p. 85). However, while the standard k-? model has proved extremely useful, it does have its limits, particularly when it comes to modeling "wake flows, buoyancy, Coriolis, curved flows and other effects" (Stangroom 2004, p. 85). As a result, researchers have developed alternatives that can more accurately model the kind of effects that will be most important for a study of high-rise cross sections. The first of these modified models is the k-? RNG model, so named because "it is based on renormalization group analysis of the Navier-Stokes equations," resulting in a slightly more complex, but ultimately more accurate computation (Stangroom 2004, p. 86). The k-? RNG model differs from the standard k-? model due to its inclusion of different constants and the combination of one constant with a function, resulting in a more accurate separation of flow and recirculation in the model.

The main benefit of the k-? RNG model over the standard model is the way that the former can more accurately model complex flows. For example, the k-? RNG can produce accurate turbulence models for flows over complex terrain with a variety of recirculation patterns, something that the standard model is simply ill-equipped to deal with (Stangroom 2004, p. 87). Furthermore, the increase in computational cost is minimal when compared to the increase in accuracy, making the trade-off well worth it in most situations. However, because the k-? RNG is "still based on the isotropic eddy viscosity concept," it will not necessarily produce better results in all situations, and in some cases, it can actually reduce accuracy (Stangroom 2004, pp. 86-87).

The main issue effecting k-? models tends to be "the overestimation of turbulent kinetic energy," meaning an overestimation of the value for k in any given instance, leading to inaccurate results (Stangroom 2004, p. 90). Situation-specific alterations to the k-? model have proved effective while simultaneously demonstrating the fragility of the model, or at least the sensitivity of the model to inaccurate or variable values. For example, while one modification accurately modeled "the flow over urban canopies," it simultaneously demonstrated that flow is extremely susceptible to obstructions such as buildings, meaning that this model would possibly need adjusting for every application, due to the extreme variability in any given building's surroundings (Stangroom 2004, p. 90).

References

Azad, R.S. 1993, the Atmospheric Boundary Layer for Engineers, Kluwer Academic

Publishers, Berlin.

Garratt, J.R. 1992, the Atmospheric Boundary Layer, Cambridge University Press, Cambridge.

Huang, S., Li, Q.S. & Xu, S. 2007, "Numerical evaluation of wind effects on a tall steel building by CFD," Journal of Constructional Steel Research, vol. 63, no.…