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Vector and Tensor Calculus
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
Scalars are completely described by their magnitude or absolute value, and they do not require direction in space for their specification. In most cases, we shall denote scalars by lower case, lightface italic type, such as p for pressure and ρ for density. Operations with scalars, i.e., addition and multiplication, follow the rules of elementary algebra. A scalar field is a real-valued function that associates a scalar (i.e., a real number) with each point of a given region in space. Let us consider, for example, the right-handed Cartesian coordinate system of Fig. 1.1 and a closed three-dimensional region V occupied by a certain amount of a moving fluid at a given time instance t. The density ρ of the fluid at any point (x, y, z) of V defines a scalar field denoted by ρ(x, y, z). If the density is, in addition, time-dependent, one may write ρ= ρ(x, y, z, t).
Differential Calculus of Scalar and Vector Fields
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
Let D be a domain in the three-dimensional space, and let f be a scalar function defined in D. Then at each point P: (x, y, z) in D, f assigns a unique real number f(x, y, z) [or f(P) for brevity], which is the value of f at that point. The domain D, together with the corresponding values of f at each point in D, is called a scalar field. The function f is said to define the scalar field in D. Oftentimes, however, when the domain is clear from the context, the scalar field is simply identified with the scalar function. Thus, for example, if at each, point of the atmosphere there is assigned a real number T(P) which represents the temperature at P, then T is a scalar field (also called a temperature field). Other examples of scalar fields are the density of air in the atmosphere, the pressure in a body of fluid, and the gravitational potential in space.
Scalar Visualization
Published in Alexandru Telea, Data Visualization, 2014
In the previous section, we described how 2D scalar fields can be visualized using the simple colormap technique. We also discussed how using too few colors in a colormap leads to undesired color-banding effects (see Figure 5.6). However, color banding is related to fundamental and widely used visualization technique called contouring. To understand contouring, think of the meaning of the sharp color transitions that separate the color bands in Figure 5.7(a).
Modelling of inflow-conditions for vortex particle methods to simulate atmospheric turbulence and its induced aerodynamic admittance on line-like bluff bodies
Published in International Journal of Computational Fluid Dynamics, 2018
Khaled Ibrahim Tolba, Guido Morgenthal
The pressure term vanishes, because the curl of the gradient of any scalar field is always zero.The second term in the left-hand side, i.e. the curl of the N–S velocity convection term; can be rewritten as, This is derived from the vector identity: The second term in the right-hand side, i.e. the curl of the N–S velocity diffusion term; can be rewritten as, This is derived from the vector identity for the Laplacian:, as well as Equation (2) due to the incompressibility condition.The previous diffusion term can be further simplified, where This is derived from the vector identity: and the fact that the divergence of the vorticity ω is always zero.
A hybrid solution to parallel calculation of augmented join trees of scalar fields in any dimension
Published in Computer-Aided Design and Applications, 2018
Paul Rosen, Junyi Tu, Les A. Piegl
In CAD applications, scalar fields are used to describe a variety of details from photographs, to laser scans, to x-ray, CT or MRI of machine parts. These scalar fields are invaluable for a variety of tasks, such as fatigue detection in parts. However, analyzing scalar fields can be quite challenging due to their size, complexity, and the need to understand both local details and global context. Augmented join trees are the key data structure used in the computation of merge trees, split trees, and contour trees [2]. By capturing geometric properties, including local minima, local maxima, and saddle points (Fig. 1), these trees are useful in the evaluation and simplification of scalar field data. This is useful for tasks such as hierarchical visualization [1,3], segmentation [5,9], or tracing structures [10] in scalar field data. However, computing these trees is expensive, and their incremental construction makes parallel computation nontrivial.
Transparency study of architectural space based on a scalar field function
Published in Spatial Cognition & Computation, 2023
Xiaofeng Lou, Kaihuai Deng, Yidi Li, ChangHai Peng
The phenomenal transparency of architectural space (PTAS) is actually a visual Scalar Field, or a Quantity Field that is perceived through vision and then imagined in the brain. So, like other physical fields that have only magnitude and no direction, such as temperature and density, it can be studied with Scalar Field Functions (SFF). The SFF is a function that assigns each point in space a scalar value – a real or physical quantity. Based on the SFF, this paper quantitatively researches the necessary and sufficient conditions, and the optimal rule for the phenomenal transparency of architecture space, as well as its design method.