scholarly journals Solving Burgers’ equation with quantum computing

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Furkan Oz ◽  
Rohit K. S. S. Vuppala ◽  
Kursat Kara ◽  
Frank Gaitan

AbstractComputational fluid dynamics (CFD) simulations are a vital part of the design process in the aerospace industry. Although reliable CFD results can be obtained with turbulence models, direct numerical simulation of complex bodies in three spatial dimensions (3D) is impracticable due to the massive amount of computational elements. For instance, a 3D direct numerical simulation of a turbulent boundary-layer over the wing of a commercial jetliner that resolves all relevant length scales using a serial CFD solver on a modern digital computer would take approximately 750 million years or roughly 20% of the earth’s age. Over the past 25 years, quantum computers have become the object of great interest worldwide as powerful quantum algorithms have been constructed for several important, computationally challenging problems that provide enormous speed-up over the best-known classical algorithms. In this paper, we adapt a recently introduced quantum algorithm for partial differential equations to Burgers’ equation and develop a quantum CFD solver that determines its solutions. We used our quantum CFD solver to verify the quantum Burgers’ equation algorithm to find the flow solution when a shockwave is and is not present. The quantum simulation results were compared to: (i) an exact analytical solution for a flow without a shockwave; and (ii) the results of a classical CFD solver for flows with and without a shockwave. Excellent agreement was found in both cases, and the error of the quantum CFD solver was comparable to that of the classical CFD solver.

Author(s):  
Alessandro Chiarini ◽  
Maurizio Quadrio

AbstractA direct numerical simulation (DNS) of the incompressible flow around a rectangular cylinder with chord-to-thickness ratio 5:1 (also known as the BARC benchmark) is presented. The work replicates the first DNS of this kind recently presented by Cimarelli et al. (J Wind Eng Ind Aerodyn 174:39–495, 2018), and intends to contribute to a solid numerical benchmark, albeit at a relatively low value of the Reynolds number. The study differentiates from previous work by using an in-house finite-differences solver instead of the finite-volumes toolbox OpenFOAM, and by employing finer spatial discretization and longer temporal average. The main features of the flow are described, and quantitative differences with the existing results are highlighted. The complete set of terms appearing in the budget equation for the components of the Reynolds stress tensor is provided for the first time. The different regions of the flow where production, redistribution and dissipation of each component take place are identified, and the anisotropic and inhomogeneous nature of the flow is discussed. Such information is valuable for the verification and fine-tuning of turbulence models in this complex separating and reattaching flow.


Author(s):  
Frank Muldoon ◽  
Sumanta Acharya

Direct Numerical Simulation (DNS) of a film cooling jet is presented. In DNS no turbulence models are introduced, and the turbulent length scales in the flow field are fully resolved. Therefore the calculations are expected to provide an accurate representation of reality, and the numerical data can be used to understand the flow physics and to compute turbulence budgets. In this paper, a DNS for an inclined jet at a jet Reynolds number of 3068 is presented. Statistics for the various budgets in the turbulence kinetic energy and dissipation rate equations are computed and presented to provide a basis for improvements to the turbulence models. A new wall function based on DNS results for a film cooling flow is presented.


Author(s):  
Naveen Pillai ◽  
Nicholas Sponsel ◽  
Katharina Stapelmann ◽  
Igor A Bolotnov

Abstract Direct Numerical Simulation (DNS) is often used to uncover and highlight physical phenomena that are not properly resolved using other Computational Fluid Dynamics (CFD) methods due to shortcuts taken in the latter to cheapen computational cost. In this work we use DNS along with interface tracking to take an in-depth look at bubble formation, departure, and ascent through water. To form the bubbles air is injected through a novel orifice geometry not unlike that of a flute submerged underwater, which introduces phenomena that are not typically brought to light in conventional orifice studies. For example, our single-phase simulations show a significant leaning effect wherein pressure accumulating at the trailing nozzle edges leads to asymmetric discharge through the nozzle hole, and an upward bias in the flow in the rest of the pipe. In our two-phase simulations, this effect is masked by the surface tension of the bubble sitting on the nozzle, but it can still be seen following departure events. After bubble departure, we observe the bubbles converge towards an ellipsoidal shape, which has been validated by experiments. As the bubbles rise, we note that local variations in the vertical velocity cause the bubble edges to flap slightly, oscillating between relatively low and high velocities at the edges. Thus, causing the bubble edges to periodically lag and lead the bulk bubble mass.


Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca

In this chapter we examine one of two main classes of algorithms: quantum algorithms that solve problems with a complexity that is superpolynomially less than the complexity of the best-known classical algorithm for the same problem. That is, the complexity of the best-known classical algorithm cannot be bounded above by any polynomial in the complexity of the quantum algorithm. The algorithms we will detail all make use of the quantum Fourier transform (QFT). We start off the chapter by studying the problem of quantum phase estimation, which leads us naturally to the QFT. Section 7.1 also looks at using the QFT to find the period of periodic states, and introduces some elementary number theory that is needed in order to post-process the quantum algorithm. In Section 7.2, we apply phase estimation in order to estimate eigenvalues of unitary operators. Then in Section 7.3, we apply the eigenvalue estimation algorithm in order to derive the quantum factoring algorithm, and in Section 7.4 to solve the discrete logarithm problem. In Section 7.5, we introduce the hidden subgroup problem which encompasses both the order finding and discrete logarithm problem as well as many others. This chapter by no means exhaustively covers the quantum algorithms that are superpolynomially faster than any known classical algorithm, but it does cover the most well-known such algorithms. In Section 7.6, we briefly discuss other quantum algorithms that appear to provide a superpolynomial advantage. To introduce the idea of phase estimation, we begin by noting that the final Hadamard gate in the Deutsch algorithm, and the Deutsch–Jozsa algorithm, was used to get at information encoded in the relative phases of a state. The Hadamard gate is self-inverse and thus does the opposite as well, namely it can be used to encode information into the phases. To make this concrete, first consider H acting on the basis state |x⟩ (where x ∊ {0, 1}). It is easy to see that You can think about the Hadamard gate as having encoded information about the value of x into the relative phases between the basis states |0⟩ and |1⟩.


2007 ◽  
Vol 7 (1&2) ◽  
pp. 83-92
Author(s):  
R. Schutzhold ◽  
W.G. Unruh

The fastest quantum algorithms (for the solution of classical computational tasks) known so far are basically variations of the hidden subgroup problem with {$f(U[x])=f(x)$}. Following a discussion regarding which tasks might be solved efficiently by quantum computers, it will be demonstrated by means of a simple example, that the detection of more general hidden (two-point) symmetries {$V\{f(x),f(U[x])\}=0$} by a quantum algorithm can also admit an exponential speed-up. E.g., one member of this class of symmetries {$V\{f(x),f(U[x])\}=0$} is discrete self-similarity (or discrete scale invariance).


2021 ◽  
Vol 24 (3) ◽  
pp. 207-221
Author(s):  
Kamil Khadiev ◽  
Vladislav Remidovskii

We study algorithms for solving the problem of assembling a text (long string) from a dictionary (a sequence of small strings). The problem has an application in bioinformatics and has a connection with the sequence assembly method for reconstructing a long deoxyribonucleic-acid (DNA) sequence from small fragments. The problem is assembling a string t of length n from strings s1,...,sm. Firstly, we provide a classical (randomized) algorithm with running time Õ(nL0.5 + L) where L is the sum of lengths of s1,...,sm. Secondly, we provide a quantum algorithm with running time Õ(nL0.25 + √mL). Thirdly, we show the lower bound for a classical (randomized or deterministic) algorithm that is Ω(n+L). So, we obtain the quadratic quantum speed-up with respect to the parameter L; and our quantum algorithm have smaller running time comparing to any classical (randomized or deterministic) algorithm in the case of non-constant length of strings in the dictionary.


2016 ◽  
Vol 139 (2) ◽  
Author(s):  
Luca Marocco ◽  
Andrea Franco

A turbulent convective flow of an incompressible fluid inside a staggered ribbed channel with high blockage at ReH ≈ 4200 is simulated with direct numerical simulation (DNS) and Reynolds-averaged Navier–Stokes (RANS) techniques. The DNS results provide the reference solution for comparison of the RANS turbulence models. The k–ε realizable, k–ω SST, and v2¯–f model are accurately analyzed for their strengths and weaknesses in predicting the flow and temperature field for this geometry. These three models have been extensively used in literature to simulate this configuration and boundary conditions but with discordant conclusions upon their performance. The v2¯–f model performs much better than the k–ε realizable while the k–ω SST model results to be inadequate.


Author(s):  
Jianxin Chen ◽  
Andrew M. Childs ◽  
Shih-Han Hung

How many quantum queries are required to determine the coefficients of a degree- d polynomial in n variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields F q , R and C . We show that k C and 2 k C queries suffice to achieve probability 1 for C and R , respectively, where k C = ⌈ ( 1 / ( n + 1 ) ) ( n + d d ) ⌉ except for d =2 and four other special cases. For F q , we show that ⌈( d /( n + d ))( n + d d ) ⌉ queries suffice to achieve probability approaching 1 for large field order q . The classical query complexity of this problem is ( n + d d ) , so our result provides a speed-up by a factor of n +1, ( n +1)/2 and ( n + d )/ d for C , R and F q , respectively. Thus, we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of 2. For the case of F q , we conjecture that 2 k C queries also suffice to achieve probability approaching 1 for large field order q , although we leave this as an open problem.


2010 ◽  
Vol 650 ◽  
pp. 307-318 ◽  
Author(s):  
JOHAN OHLSSON ◽  
PHILIPP SCHLATTER ◽  
PAUL F. FISCHER ◽  
DAN S. HENNINGSON

A direct numerical simulation (DNS) of turbulent flow in a three-dimensional diffuser at Re = 10000 (based on bulk velocity and inflow-duct height) was performed with a massively parallel high-order spectral element method running on up to 32768 processors. Accurate inflow condition is ensured through unsteady trip forcing and a long development section. Mean flow results are in good agreement with experimental data by Cherry et al. (Intl J. Heat Fluid Flow, vol. 29, 2008, pp. 803–811), in particular the separated region starting from one corner and gradually spreading to the top expanding diffuser wall. It is found that the corner vortices induced by the secondary flow in the duct persist into the diffuser, where they give rise to a dominant low-speed streak, due to a similar mechanism as the ‘lift-up effect’ in transitional shear flows, thus governing the separation behaviour. Well-resolved simulations of complex turbulent flows are thus possible even at realistic Reynolds numbers, providing accurate and detailed information about the flow physics. The available Reynolds stress budgets provide valuable references for future development of turbulence models.


Hydrology ◽  
2021 ◽  
Vol 8 (4) ◽  
pp. 151
Author(s):  
Mehdi Heyrani ◽  
Abdolmajid Mohammadian ◽  
Ioan Nistor

This study uses a computational fluid dynamics (CFD) approach to simulate flows in Parshall flumes, which are used to measure flowrates in channels. The numerical results are compared with the experimental data, which show that choosing the right turbulence model, e.g., v2−f and LC, is the key element in accurately simulating Parshall flumes. The Standard Error of Estimate (SEE) values were very low, i.e., 0.76% and 1.00%, respectively, for the two models mentioned above. The Parshall flume used for this experiment is a good example of a hydraulic structure for which the design can be more improved by implementing a CFD approach compared with a laboratory (physical) modeling approach, which is often costly and time-consuming.


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