Quantum computing promises a paradigm shift in our computational capability, and among the most promising applications is efficiently solving partial differential equations. Quantum computers gain their advantage by storing and processing information in a space that grows exponentially with the size of the computer, using the principles of quantum mechanics. A quantum algorithm is proposed for solving the advection equation using sparse Hamiltonian simulation. Discretising the advection equation using the finite difference method with explicit Euler time integration allows a time step to be expressed as a matrix transformation on a vector, $\vec{\phi}_{t+1} = A\vec{\phi}_{t}$. This nonunitary matrix is embedded within a Hamiltonian to evolve the quantum state according to a unitary operator that solves the corresponding Schr\"{o}dinger equation. The Hamiltonian embedding procedure ensures that a portion of the statevector evolves according to the advection equation. The unitary operator accurately embeds the matrix even for large Hamiltoninan evolution times, enabling a time step to be optimised to achieve a near-certain success rate with errors comparable to those of the Euler method. Qubit requirements grow logarithmically with the number of grid points $N$ due to the exponentially growing capacity to store information. The required circuit depth grows polynomially as $\widetilde{O}(N^{1/3}k/\epsilon)$ for a three-dimensional problem, where $k$ and $\epsilon$ are the order of spatial discretisation and the allowable error respectively. This offers a significant polynomial speedup over classical algorithms with a complexity of approximately $O(N^{4/3})$. The algorithm has been implemented using statevector simulations to calculate the evolution of a scalar in a laminar, two-dimensional channel flow using a combination of Dirichlet and periodic boundary conditions. The velocity is known to be a parabolic profile between the two plates, so can be encoded into the matrix $A$ prior to the simulation. The results confirm the analytical findings that the statevector accurately evolves according to the advection equation, and comparisons with classical computations show that the scalar field deviates by no more than 1\% throughout the simulation with a typical Courant-Friedrichs-Lewy number of 0.1. The results represent a promising option for performing computational fluid dynamics on a quantum computer.