scholarly journals Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

Computation ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 48 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Fixed Point ◽  
Relative Error ◽  
Symbolic Regression ◽  
Logarithmic Function ◽  
Flow Friction ◽  
First Order ◽  
Quasi Monte Carlo ◽  
Initial Stage ◽  
Starting Point ◽  
Transcendental Functions

The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.

scholarly journals Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Fixed Point ◽  
Friction Factor ◽  
Fixed Point Method ◽  
Point Method ◽  
Symbolic Form ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Starting Point ◽  
Derivatives Of

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

scholarly journals Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 26 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Rational Approximation ◽  
Numerical Experiments ◽  
The Novel ◽  
Computationally Efficient ◽  
Flow Friction ◽  
Quasi Monte Carlo ◽  
Transcendental Functions ◽  
Processor Unit ◽  
Computer Processor ◽  
Rational Expressions

The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Padé approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function.

scholarly journals Review of New Flow Friction Equations: Constructing Colebrook’s Explicit Correlations Accurately

2020 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Monte Carlo ◽  
Series Expansion ◽  
Relative Error ◽  
Asymptotic Series ◽  
Computationally Efficient ◽  
Arithmetic Operations ◽  
Flow Friction ◽  
Quasi Monte Carlo ◽  
Computationally Expensive ◽  
Maximal Relative Error

Using only a limited number of computationally expensive functions, we show a way how to construct accurate and computationally efficient approximations of the Colebrook equation for flow friction. The presented approximations are based on the asymptotic series expansion of the Wright &omega;-function and symbolic regression. The results are verified with 8 million of Quasi-Monte Carlo points covering the domain of interest for engineers. In comparison with the built-in &ldquo;wrightOmega&rdquo; feature of Matlab R2016a, the herein introduced related approximations of the Wright &omega;-function significantly accelerate the computation. With only two logarithms and several basic arithmetic operations used, the presented approximations are not only computationally efficient but also extremely accurate. The maximal relative error of the most promising approximation which is given in the form suitable for engineers&rsquo; use is limited to 0.0012%, while for a little bit more complex variant is limited to 0.000024%.

scholarly journals Chemical Oxidative Polymerization of Methylene Blue: Reaction Mechanism and Aspects of Chain Structure

Polymers ◽  
2021 ◽  
Vol 13 (13) ◽  
pp. 2188
Author(s):  
Yaroslav O. Mezhuev ◽  
Igor Y. Vorobev ◽  
Ivan V. Plyushchii ◽  
Efrem G. Krivoborodov ◽  
Alexander A. Artyukhov ◽  
...  
Keyword(s):  
Methylene Blue ◽  
Oxidative Polymerization ◽  
Chain Structure ◽  
Charge Transfer Complexes ◽  
Significant Weight Loss ◽  
Chemical Oxidative Polymerization ◽  
First Order ◽  
Initial Stage ◽  
Blue Reaction ◽  
Heating Up

The kinetic regularities of the initial stage of chemical oxidative polymerization of methylene blue under the action of ammonium peroxodisulfate in an aqueous medium have been established by the method of potentiometry. It was shown that the methylene blue polymerization mechanism includes the stages of chain initiation and growth. It was found that the rate of the initial stage of the reaction obeys the kinetic equation of the first order with the activation energy 49 kJ × mol−1. Based on the proposed mechanism of oxidative polymerization of methylene blue and the data of MALDI, EPR, and IR spectroscopy methods, the structure of the polymethylene blue chain is proposed. It has been shown that polymethylene blue has a metallic luster, and its electrical conductivity is probably the result of conjugation over extended chain sections and the formation of charge transfer complexes. It was found that polymethylene blue is resistant to heating up to a temperature of 440 K and then enters into exothermic transformations without significant weight loss. When the temperature rises above 480 K, polymethylene blue is subject to endothermic degradation and retains 75% of its mass up to 1000 K.

scholarly journals Chiral perturbation theory for GR

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Kirill Krasnov ◽  
Yuri Shtanov
Keyword(s):  
Perturbation Theory ◽  
Chiral Perturbation ◽  
Perturbative Calculation ◽  
New Formalism ◽  
Yang Mills ◽  
First Order ◽  
Starting Point ◽  
Gauge Fixing ◽  
New Perturbation ◽  
Effective Description

Abstract We describe a new perturbation theory for General Relativity, with the chiral first-order Einstein-Cartan action as the starting point. Our main result is a new gauge-fixing procedure that eliminates the connection-to-connection propagator. All other known first-order formalisms have this propagator non-zero, which significantly increases the combinatorial complexity of any perturbative calculation. In contrast, in the absence of the connection-to-connection propagator, our formalism leads to an effective description in which only the metric (or tetrad) propagates, there are only cubic and quartic vertices, but some vertex legs are special in that they cannot be connected by the propagator. The new formalism is the gravity analog of the well-known and powerful chiral description of Yang-Mills theory.

Baatɔnu Personal Names from Birth to Death

Africa ◽  
2000 ◽  
Vol 70 (1) ◽  
pp. 79-106 ◽  
Author(s):  
Wendy Schottman
Keyword(s):  
Family Relations ◽  
Personal Names ◽  
Initial Stage ◽  
Starting Point

AbstractThe circumstances of a child’s birth define his or her starting point in life, and they will be inscribed in the child’s file, so to speak, by means of a set of rule-governed birth names. These ‘child names’ are perfectly suitable for this initial stage of life, but all Baatɔmbu aspire one day to replace this original set of ‘orthodox’ names by another orthodox name, an inherited title name, corresponding to an achieved social and spiritual status. Commoners and nobles have separate institutions of gɔɔbiru, ‘inherited title names’, but in both cases the successive bearers of a gɔɔbiru share an exemplary essence that each must honour and perpetuate with his life. Baatɔnu nobles bestow baptism names on children around the age of seven, allowing these young candidates for the various gɔɔbiru to be matched, according to their potential, with a name whose influence will guide them into adulthood. Joking names and teknonyms can be classified as non-orthodox or informal names and seem to fill a gap left by the orthodox names, allowing personal and family relations to be expressed and negotiated. These names carry no prestige, but their use affords pride and pleasure and, unlike orthodox names, they can be used without infringing ‘shame’-based taboos.

scholarly journals BACH FLOWS OF PRODUCT MANIFOLDS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250039 ◽  
Author(s):  
SANJIT DAS ◽  
SAYAN KAR
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Ricci Flow ◽  
Flow Characteristics ◽  
Geometric Flow ◽  
Flow Equations ◽  
First Order ◽  
Bach Tensor ◽  
Point Condition ◽  
Future Work

We investigate various aspects of a geometric flow defined using the Bach tensor. First, using a well-known split of the Bach tensor components for (2, 2) unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like S2 × S2, R2 × S2. In addition, we obtain the fixed-point condition for general (2, 2) manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped 4-manifolds, we reduce the flow equations to a first-order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work.

TERBIUM MOLYBDATE: GROUP THEORY AND FLUCTUATIONS

1987 ◽  
Vol 01 (05n06) ◽  
pp. 239-244
Author(s):  
SERGE GALAM
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Parameter Space ◽  
Landau Theory ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Theory Analysis ◽  
Continuous Transition ◽  
Dimensional Parameter ◽  
First Order

A new mechanism to explain the first order ferroelastic—ferroelectric transition in Terbium Molybdate (TMO) is presented. From group theory analysis it is shown that in the two-dimensional parameter space ordering along either an axis or a diagonal is forbidden. These symmetry-imposed singularities are found to make the unique stable fixed point not accessible for TMO. A continuous transition even if allowed within Landau theory is thus impossible once fluctuations are included. The TMO transition is therefore always first order. This explanation is supported by experimental results.

ANOMALY FREE SUPERGRAVITY IN D=10: I) THE BIANCHI IDENTITIES AND THE BOSONIC LAGRANGIAN

1988 ◽  
Vol 03 (04) ◽  
pp. 953-1021 ◽  
Author(s):  
RICCARDO D’AURIA ◽  
PIETRO FRÉ ◽  
MARIO RACITI ◽  
FRANCO RIVA
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Variational Equation ◽  
Second Order ◽  
Effective Theory ◽  
Spin Connection ◽  
Bianchi Identities ◽  
First Order ◽  
Interaction Terms ◽  
Starting Point

Using a theorem by Bonora-Pasti and Tonin on the existence of a solution for D=10N=1 Bianchi identities in the presence of a Lorentz Chern Simons term, we find an explicit parametrization of the superspace curvatures. Our solution depends only on one free parameter which can be reabsorbed in a field redefinition of the dilaton and of the gravitello. We emphasize that the essential point which enables us to obtain a closed form for the curvature parametrizations and hence for the supersymmetry transformation rules is the use of first order formalism. The spin connection is known once the torsion is known. This latter, rather than being identified with Hµνρ as it is usually done in the literature, is related to it by a differential equation which reduces to the algebraic relation Hµνρ = - 3Tµνρ e4/3σ only at γ1=0 (γ1 being proportional to κ/g2). The solution of the Bianchi identities exhibited in this paper corresponds to a D=10 anomaly free supergravity (AFS). This theory is unique in first order formalism but corresponds to various theories in second order formalism. Indeed the torsion equation is a differential equation which, in order to be solved must be supplemented with boundary conditions. One wonders whether supplemented with a judicious choice of boundary conditions for the torsion equation, AFS yields all the interaction terms found in the effective theory of the heterotic string (ETHS). In this respect two remarks are in order. Firstly it appears that solving the torsion equation iteratively with Tµνρ = -1/3Hµνρ e-4/3σ as starting point all the terms of ETHS except those with a ζ(3) coefficient show up. (Whether the coefficient agree is still to be checked.) Secondly, as shown in this paper the rheonomic solution of the super Poincaré Bianchi identities is unique. Hence additional interaction terms can be added to the Lagrangian only by modifying the rheonomic parametrization of the [Formula: see text]-curvature. The only assumption made in our paper is that [Formula: see text] has at most ψ∧ψ∧V components (sector (1,2)). Correspondingly the only room left for a modification of the present theory is the addition of a (0, 3) part in the rheonomic parametrization of [Formula: see text]. When this work was already finished a conjecture was published by Lechner Pasti and Tonin that such a generalization of AFS might exist and be responsible for the ζ(3) missing term. Indeed if we were able to solve the [Formula: see text]-Bianchi with this new (0, 3)-part then the torsion equation would be modified via new terms which, in second order formalism, lead to additional gravitational interactions. The equation of motion of Anomaly Free Supergravity can be worked out from the Bianchi identities: we indicate through which steps. The corresponding Lagrangian could be constructed with the standard procedures of the rheonomy approach. In this paper we limit ourselves to the bosonic sector of such a Lagrangian and we show that it can indeed be constructed in such a way as to produce the relation between Hµνρ and Tµνρ as a variational equation.

μ-definable sets of integers

1993 ◽  
Vol 58 (1) ◽  
pp. 291-313 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Fixed Point ◽  
Free Variable ◽  
Order Logic ◽  
First Order Logic ◽  
Inductive Definition ◽  
Natural Class ◽  
First Order ◽  
Inductive Definitions ◽  
Definable Sets ◽  
The Universe

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.

Sign in / Sign up

Export Citation Format

Share Document